Relativization of dynamical properties

SCIENCE CHINAMathematics

. REVIEWS . May 2012 Vol. 55 No. 5: 913–936

doi: 10.1007/s11425-011-4332-4

c© Science China Press and Springer-Verlag Berlin Heidelberg 2011 math.scichina.com www.springerlink.com

Relativization of dynamical properties

ZHANG GuoHua1,2

1School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433, China;2School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

Email: [email protected]

Received January 12, 2011; accepted July 21, 2011; published online December 5, 2011

Abstract In the past twenty years, great achievements have been made by many researchers in the studies of

chaotic behavior and local entropy theory of dynamical systems. Most of the results have been generalized to

the relative case in the sense of a given factor map. In this survey we offer an overview of these developments.

Keywords relative entropy, asymptotic pair, scrambled set, local variational principles, relative entropy

tuple, relative topological Pinsker factor, relative UPE, relative CPE

MSC(2010) 37A05, 37A35

Citation: Zhang G H. Relativization of dynamical properties. Sci China Math, 2012, 55(5): 913–936, doi: 10.1007/

s11425-011-4332-4

1 Introduction

Dynamical system theory is the study of the qualitative properties of group actions on spaces with

certain structures. From the viewpoint of topological dynamics, we are mostly interested in actions by

homeomorphisms on compact metric spaces with an additional structure of a Borel probability measure

invariant under the actions. In this survey, we only consider Z-actions.

Throughout the whole survey, by a topological dynamical system (TDS) we mean a pair (X,T ), where

X is a compact metric space and T : X → X is a homeomorphism; by a measure-theoretical dynamical

system (MDS) we mean a quadruple (X,B, μ, T ), where (X,B, μ) is a Lebesgue space and T : (X,B, μ) →(X,B, μ) is an invertible measure-preserving transformation. It is well known that, associated to a TDS

(X,T ), there are always invariant Borel probability measures. Let μ be such a measure over (X,T ). Then

(X,BμX , μ, T ) is an MDS, where BX is the Borel σ-algebra of X and Bμ

X is the completion of BX under

μ (sometimes we also denote (X,BμX , μ, T ) by (X,BX , μ, T ) if there is no ambiguity). So ergodic theory

involves the study of properties of a TDS.

Since the discovery of Lorenz’s attractor in the 1960s, the study of chaos theory has played a key role in

nonlinear science. In 1975 Li and Yorke [58] introduced Li-Yorke pairs and Li-Yorke chaos in a tentative

definition of chaos. The Li-Yorke definition of chaos has proved its value for interval maps. Today there

are various definitions of what it means for a map to be chaotic. A natural question arises immediately

in topological dynamics about what topological properties imply chaotic behavior.

Ergodic theory and topological dynamics exhibit a remarkable parallelism. For example, we speak

about measure-theoretic entropy in ergodic theory and topological entropy in topological dynamics. Even

if dynamical entropy is not something one can measure physically, positive entropy is often considered to

indicate strong chaotic features in a dynamical system modelling some phenomena.

914 Zhang G H Sci China Math May 2012 Vol. 55 No. 5

The first progress was made in [41] by proving that a transitive non-periodic system with a periodic

point is Li-Yorke chaotic, thus solving an open question whether Devaney chaos implies Li-Yorke chaos.

Later another long standing open question was solved in [9] by proving that any TDS with positive

topological entropy is Li-Yorke chaotic. At the same time, it was proved in [11] that there is a measure-

theoretically ‘rather big’ set of proper asymptotic pairs for any TDS with positive topological entropy.

The class of Kolmogorov systems in ergodic theory is an important class and such systems completely

differ from an MDS with zero measure-theoretic entropy. Starting with the study of seeking the coun-

terpart of a Kolmogorov system in topological dynamics, since the 1990s much attention has been paid

to the so called local properties of entropy and its many interesting results. For a survey of the whole

theory, see [33].

To get a topological analogy of the measure Kolmogorov system, Blanchard introduced the notions of

complete positive entropy (CPE) and uniform positive entropy (UPE) in topological dynamics (see [6]).

He then introduced topological entropy pairs as markers of positive topological entropy and used this to

show that any UPE TDS is disjoint from all minimal TDSs with zero topological entropy (see [7]). Later

on, in [10] the authors were able to define entropy pairs for an invariant measure and showed that for

each invariant measure, the set of entropy pairs for this measure is contained in the set of all topological

entropy pairs. Blanchard, Glasner and Host proved that the converse of [10] is also valid (see [8]), that

is, there is an invariant measure such that the set of all topological entropy pairs is contained in the set

of entropy pairs for this measure. A characterization of the set of entropy pairs for an invariant measure

as the support of some measure was obtained by Glasner (see [27]). But, why should all the information

on entropy be restricted to pairs? To obtain a better understanding of the topological version of a

Kolmogorov system, in [43] Huang and Ye introduced the notions of entropy n-tuples (where n ∈ N\{1})both in topological and measure-theoretical settings, established the variational relation between these two

kinds of entropy tuples and proved that the dynamical behaviors in any neighborhood of a topological

entropy n-tuple are very complicated. Just recently, those notions (of entropy pairs and tuples) were

generalized to entropy sequences, entropy sets and entropy points in both settings (see [12, 22, 77]).

Just note that, in order to establish the variational relationship between these two kinds of entropy

pairs and tuples, the classical variational principle was localized into local ones by many researchers in

[8,32,43,44,69]. For other related results see [2,13,20,21,28–31,38–40,45,46,50–52,56,60,67,70,71,80,81].

Let (X,T ) and (Y, S) be TDSs. By a factor map we mean π : (X,T ) → (Y, S), where π : X → Y is a

continuous surjection satisfying π ◦ T = S ◦ π. In this case, (X,T ) is called an extension of (Y, S) and

(Y, S) a factor of (X,T ). For each n ∈ N\{1}, set

R(n)π = {(x1, . . . , xn) ∈ Xn : π(x1) = · · · = π(xn)}.

Given a factor map between TDSs, one may also talk about the notions of relative topological entropy

and relative measure-theoretic entropy. Thus it is a very natural question whether the above mentioned

results can be generalized to the relative case in the sense of a given factor map between TDSs. This

question was addressed by many researchers in [31, 44, 45, 56, 67, 78, 79] and the references therein.

This survey tries to give an overview of these advances in the relative case.

First, by characterizing the relative Pinsker σ-algebra for a given TDS associated with an invariant

Borel probability measure and an invariant sub-σ-algebra, in [78] the author proved that, for a given

factor map between TDSs with positive relative topological entropy, the Li-Yorke chaos may happen on

the fibers of the map. In fact, the scrambled set on the fibers may be very ‘big’ in the sense of topological

entropy. Additionally, some other types of Li-Yorke chaos on the fibers of a factor map were discussed

in [79]. Section 3 of the survey aims to introduce these results. In Section 4, we discuss the asymptotic

behavior on the fibers of a factor map with positive relative topological entropy, most of which was also

done in [78]. Given a factor map between TDSs, note that we can talk about not only relative topological

entropy but also relative measure-theoretic entropy, and there is also a variational principle concerning

them. As, in the process of building the local entropy theory, the classical variational principle was

Zhang G H Sci China Math May 2012 Vol. 55 No. 5 915

localized into the local ones, it seems possible to establish some local variational principles concerning

relative entropy, which was at last finished in [44]. Thus, in Section 5 we will tell this story. Section 6

introduces some applications of the local variational principle concerning relative entropy, by introducing

relative entropy tuples both in topological and measure-theoretical settings and building the variational

relationship between them. Based on the relative topological pairs introduces in the last section we

discuss the relative topological Pinsker factor, relative UPE and CPE extensions. The results in the last

two sections were studied systematically in [45].

We should note that the study of relativization of dynamical properties is not a trivial generalization of

the absolute case. On one hand, by letting the factor be trivial (i.e., a singleton) the dynamical properties

in the setting of the absolute case follow directly from those of the relative setting. On the other hand,

since the publication of the pioneering paper [63] by Ornstein and Weiss in 1987, more and more people

have been paying much attention to the study of dynamical properties of an amenable group action

(the definition of the amenability of a group is omitted here, we only emphasize that for a dynamical

system of group actions, the amenability of the group ensures the existence of invariant Borel probability

measures under the group actions. The class of amenable groups includes all finite groups, solvable groups

and compact groups). The study of relativization of dynamical properties (especially the local relative

entropy theory) served as a bridge and played a very important role in the study of local entropy theory

of dynamical system of an amenable group action (see [46]) and in the study of local entropy theory of a

continuous bundle random dynamical system (see [21]).

2 Preliminaries

In this section, we give a brief introduction of concepts used later.

2.1 Relative measure-theoretic entropy

Let (X,BX , T, μ) and (Y,BY , S, ν) be two MDSs. A given measurable map

π : (X,BX , T, μ) → (Y,BY , S, ν)

is called a homomorphism if πμ = ν and π ◦ T = S ◦ π (and then (Y,BY , S, ν) is called a factor of

(X,BX , T, μ)). In this case, π is called an isomorphism if additionally it is invertible and π−1 is also

a measurable measure-preserving transformation. In order to distinguish between two MDSs which are

spectrally isomorphic, in 1958 Kolmogorov introduced an isomorphism invariant which is called measure-

theoretic entropy (see [53]).

Let (X,B, μ, T ) be an MDS. A family of subsets of X with union X is called a partition if all elements

of the family are disjoint. Moreover, all partitions of X are assumed to consist of atoms belonging to B.Denote by PX the collection of all finite partitions of X . For α, β ∈ PX , α is said to be finer than β

(write β � α or α � β) if each atom of α is contained in some atom of β. Let

α ∨ β = {A ∩B : A ∈ α,B ∈ β}.

It works similarly for any countable family in PX . Given a partition P of X and x ∈ X , denote by P (x)

the atom of P containing x. If {Pi : i ∈ I} ⊆ PX is a countable family, the partition P =∨

i∈I Pi is

called a measurable partition. The collection of all sets B ∈ B, which is the union of atoms of P , form a

sub-σ-algebra of the σ-algebra B denoted by P ∗. Let P1, P2 be two measurable partitions of X , P ∗1 ∨ P ∗

2

denotes the smallest sub-σ-algebra of B containing the σ-algebras P ∗1 and P ∗

2 , it can be shown that

P ∗1 ∨ P ∗

2 = (P1 ∨ P2)∗,


and so there is no ambiguity in denoting P ∗ by P . By [68], every sub-σ-algebra of B coincides with a

σ-algebra constructed in this way outside a set of measure zero. For a given measurable partition P , put

P− =

+∞∨

n=1

T−nP and PT =

+∞∨

n=−∞T−nP.

Let (X,B, μ, T ) be an MDS. Given a T -invariant sub-σ-algebra A and α ∈ PX , set

Hμ(α | A) =∑

A∈α

∫

X

−E(1A | A)(x) logE(1A | A)(x)dμ(x),

where E(1A | A) is the μ-expectation of 1A with respect to A. As A is T -invariant, it is easy to check that

the sequence Hμ(∨n−1

i=0 T−iα | A) is sub-additive and then we can define the relative measure-theoretic

μ-entropy of α with respect to A by

hμ(T, α | A) = limn→+∞

1

nHμ

( n−1∨

i=0

T−iα | A)

= infn�1

1

nHμ

( n−1∨

i=0

T−iα | A)

(= Hμ(α | α− ∨ A)),

where the last identity is not hard to obtain. Now, we can define the relative measure-theoretic μ-entropy

of (X,T ) with respect to A by

hμ(T,X | A) = supα∈PX

hμ(T, α | A).

In the case of A being trivial, i.e., A = {∅, X}, we have directly that

Hμ(α | {∅, X}) = −∑

A∈α

μ(A) log μ(A) (denoted by Hμ(α)),

and write

hμ(T, α) = hμ(T, α | {∅, X}), hμ(T,X) = hμ(T,X | {∅, X})(called the measure-theoretic μ-entropy of α and (X,T ), respectively).

Now let π : (X,BX , μ, T ) → (Y,BY , ν, S) be a homomorphism between MDSs. Clearly, π−1BY is a

T -invariant sub-σ-algebra of BX , and so we denote

hμ(T, α | π) = hμ(T, α | π−1BY )

for each α ∈ PX and

hμ(T,X | π) = hμ(T,X | π−1BY ).

2.2 (Relative) Topological entropy

In 1965, Adler, Konheim and McAndrew introduced the concept of topological entropy for TDSs (see [1]),

and then in his 1971 paper [14] Bowen introduced his definition of topological entropy for a subset of a

given TDS using his separated and spanning sets (see also [19, 74]).

Let (X,T ) be a TDS with d a compatible metric on the space. For each n ∈ N we define a new metric

dn on the space X by

dn(x, y) = max0�i�n−1

d(T ix, T iy).

Let ε > 0 and K ⊆ X . A subset F of X is said to be (n, ε) span K with respect to T if for each x ∈ K,

there exists y ∈ F with dn(x, y) � ε, and a subset E of K is said to be (n, ε) separated with respect

to T if x, y ∈ E, x �= y implies dn(x, y) > ε. Let rn(d, T, ε,K) denote the smallest cardinality of any


(n, ε)-spanning set for K with respect to T and sn(d, T, ε,K) denote the largest cardinality of any (n, ε)

separated subset of K with respect to T . Bowen’s topological d-entropy of K is then defined as

hd(T,K) = limε→0+

lim supn→+∞

1

nlog sn(d, T, ε,K)

(

= limε→0+

lim supn→+∞

1

nlog rn(d, T, ε,K)

)

,

where the last identity is not hard to obtain.

Let (X,T ) be a TDS. A family of subsets of X with union X is called a cover of X . Moreover, all

covers of X are assumed to consist of atoms belonging to the Borel σ-algebra BX . Denote by CX the

collection of all finite open covers of X . Similarly, for covers U and V , U is said to be finer than V (write

V � U or U � V) if each atom of U is contained in some atom of V .Now let π : (X,T ) → (Y, S) be a factor map between TDSs and U ∈ CX . For any E ⊆ X , denote by

N(U , E) the minimal cardinality of any sub-cover of U which covers E (set N(U , ∅) = 1 by convention).

Let

N(U | π) = supy∈Y

N(U , π−1(y)).

As π is a factor map, it is not hard to see that the sequence logN(∨n−1

i=0 T−iU | π) is sub-additive and so

we can define the relative topological entropy of U relevant to π by

htop(T,U | π) = limn→+∞

1

nlogN

( n−1∨

i=0

T−iU | π)

= infn�1

1

nlogN

( n−1∨

i=0

T−iU | π)

.

Then the relative topological entropy of (X,T ) relevant to π is defined as

htop(T,X | π) = supU∈CX

htop(T,U | π).

Similarly, in the case of π or Y being trivial, i.e., Y is a singleton, we write

htop(T,U) = htop(T,U | π), htop(T,X) = htop(T,X | π)

(called the topological entropy of U and (X,T ), respectively). We also should note that it is not hard to

obtain

hd(T,K) = supU∈CX

lim supn→+∞

1

nlogN

( n−1∨

i=0

T−iU ,K)

,

thus, Bowen’s topological d-entropy of K is independent of the selection of the compatible metric d on

the space X , and so it is called the topological entropy of K.

2.3 Variational principle concerning (relative) entropy

Let (X,T ) be a TDS. Denote by M(X) the set of all Borel probability measures on X , M(X,T ) ⊆ M(X)

the set of all T -invariant elements, and Me(X,T ) ⊆ M(X,T ) the set of all ergodic elements. Then

both M(X) and M(X,T ) are convex compact metric spaces endowed with the weak star topology, and

Me(X,T ) �= ∅.Since the introduction of measure-theoretic entropy by [53] and topological entropy by [1], the re-

lationship between these two kinds of entropy has gained a lot of attention. In his 1969 paper [35],

Goodwyn showed that hμ(T,X) � htop(T,X) for each μ ∈ M(X,T ) and later in [34], Goodman proved

suphν(T,X) � htop(T,X), where the supremum is taken over all elements ν from M(X,T ), completing


the classical variational principle. For a short proof of it, see [61]. Later, the classical variational princi-

ple was extended to the relative case (see [15, 23, 57]). That is, let π : (X,T ) → (Y, S) be a factor map

between TDSs. Then the following relative variational principle holds:

supμ∈M(X,T )

hμ(T,X | π) = supμ∈Me(X,T )

hμ(T,X | π) = htop(T,X | π)

= supy∈Y

htop(T, π−1(y)).

2.4 Pinsker σ-algebra and Kolmogorov systems

Once measure-theoretic entropy is defined for MDSs, two subclasses immediately stand out: zero-entropy

systems and Kolmogorov systems. Kolmogorov systems play an important role in ergodic theory, and

can be defined as follows. An MDS (X,B, μ, T ) is called a Kolmogorov system if one of the following

equivalent properties holds:

(1) hμ(T, α) > 0 once α ∈ PX consists of two non-trivial elements.

(2) hμ(T, α) > 0 for each non-trivial α ∈ PX , i.e., α is not the trivial partition {X}.(3) Each non-trivial factor of (X,B, μ, T ) has positive measure-theoretic entropy.

(4) There exists a measurable partition P satisfying that

T−1P � P,

+∞∧

n=0

T−nP = {∅, X} and

+∞∨

n=0

T−nP

is just the point partition of the system (X,B, μ, T ).Let (X,B, μ, T ) be an MDS and A ⊆ B a T -invariant sub-σ-algebra. The relative Pinsker σ-algebra

Pμ(A) is defined as the smallest σ-algebra containing {ξ ∈ PX : hμ(T, ξ | A) = 0}. In the case of A being

trivial, denote it by Pμ(X,T ) (called the Pinsker σ-algebra of (X,T )). Thus, from the above equivalent

definitions, one has that (X,B, μ, T ) is a Kolmogorov system if and only if Pμ(X,T ) = {∅, X}.

Part IChaotic behavior in fibers over a given factor

3 Li-Yorke chaos in fibers over a given factor

Since the discovery of Lorenz’s attractor in the 1960s, the study of chaos theory has played a key role

in nonlinear science. Today there are various definitions of what it means for a map to be chaotic, and

some of them work reasonably only in particular phase spaces (most of them were reviewed in [54, 55]).

In 1975 Li and Yorke [58] introduced Li-Yorke pairs and Li-Yorke chaos in a tentative definition of chaos.

The Li-Yorke definition of chaos has also proved its value for interval maps.

Definition 3.1. Let (X,T ) be a TDS. (x1, x2) ∈ X2 is called a Li-Yorke pair if

lim infn→+∞ d(T nx1, T

nx2) = 0 and lim supn→+∞

d(T nx1, Tnx2) > 0.

Let ∅ �= S ⊆ X . S is called scrambled if S contains at least two different points and any pair (x1, x2)

of different points from S is a Li-Yorke pair. Then TDS (X,T ) is called Li-Yorke chaotic if there is an

uncountable scrambled subset S ⊆ X .

Given a TDS, which of its topological properties imply Li-Yorke chaos? Note that positive topological

entropy of a TDS can be interpreted as a manifestation of chaotic behavior. The first progress was made

by Huang and Ye in [41] by proving that a transitive non-periodic TDS with a periodic point is Li-Yorke

chaotic, thus solving the long-open question whether Devaney chaos implies Li-Yorke chaos. See [5,18] for


more story of Devaney’s chaos. Let (X,T ) be a TDS. Recall that (X,T ) is minimal if whenever U ⊆ X

is a non-empty open subset there exists N ∈ N satisfying⋃N

i=0 T−iU = X and is transitive if whenever

U1, U2 ⊆ X are non-empty open subsets there exists n ∈ N with T−nU1 ∩ U2 �= ∅. Obviously, each

minimal TDS is transitive. Later on, another long-standing open question was answered affirmatively

in [9] by showing that each TDS with positive topological entropy is Li-Yorke chaotic.

Theorem 3.2. Let (X,T ) be a TDS and μ ∈ Me(X,T ).

(1) If (X,BX , μ, T ) is not measure-distal, then (X,T ) is Li-Yorke chaotic.

(2) If (X,T ) has positive topological entropy, then it is Li-Yorke chaotic.

In the process of proving Theorem 3.2, the tools and methods from ergodic theory (especially, the

Furstenberg-Zimmer structure theorem for ergodic MDSs (see [24, 82, 83])) become inevitable, which

presents every ergodic MDS as a weakly mixing extension of a measure-distal MDS. Recently, in their

2007 paper [51], Kerr and Li gave a purely topological proof of this fact based on a combinatorial approach

to the study of IE-tuples (introduced in the same paper). Remark that in [43] the authors also discussed

such independence property (there it was not called an IE-tuple).

Definition 3.3. Let (X,T ) be a TDS and k ∈ N. Set

Δk(X) = {(x1, . . . , xk) ∈ Xk : x1 = · · · = xk}.(x1, . . . , xk) ∈ Xk\Δk(X) is called an IE-tuple if, for every product neighborhood U1 × · · · × Uk of it,

(U1, . . . , Uk) has an independent set J ⊆ Z+ of positive density, i.e., the limit limn→+∞|J∩{0,1,...,n}|

n+1

exists and is nonzero and⋂

i∈I T−iUσ(i) �= ∅ for every non-empty finite subset I ⊆ J and function

σ : I → {1, . . . , k}.By some combinatorial arguments, Kerr and Li [51] proved the following (remark that Proposition 3.4

(1) was firstly proved in [43]):

Proposition 3.4. Let (X,T ) be a TDS and k ∈ N\{1}.(1) If (X,T ) has positive topological entropy, then there exists (x1, . . . , xk) ∈ Xk\Δk(X) such that

(x1, . . . , xk) is an IE-tuple.

(2) If (x1, . . . , xk) ∈ Xk\Δk(X) is an IE-tuple with a product neighborhood U1×· · ·×Uk, then there exist

Cantor sets Zj ⊆ Uj, j = 1, . . . , k such that: (a) every non-empty finite tuple of points from Z.=

⋃kj=1 Zj

is an IE-tuple; and (b) for each m ∈ N and distinct y1, . . . , ym ∈ Z, y′1, . . . , y′m ∈ Z, one has

lim infn→+∞ max

1�i�kd(T nyi, y

′i) = 0.

Thus, as a direct corollary, one obtains again Theorem 3.2 (2).

In fact, it turns out that Li-Yorke chaos actually follows from local topological weak mixing (introduced

and studied recently by Blanchard and Huang [12] and Oprocha and Zhang [64–66]) rather than positive

entropy, see also [76] for some related results. Based on this, we could generalize the above results to the

relative case in the setting of a given factor map between TDSs.

Definition 3.5. Let (X,T ) be a TDS and ∅ �= A ⊆ X,n ∈ N\{1}. We say that A is

(1) transitive if for each pair of open subsets (U, V ) of X intersecting A, there exists m ∈ N with

Tm(V ∩A) ∩ U �= ∅.(2) weakly mixing of order n if An is a transitive set of (Xn, T (n)), where T (n) acts naturally on Xn

by T (n) : (x1, . . . , xn) �→ (Tx1, . . . , T xn).

(3) weakly mixing if A is weakly mixing of order m for each m ∈ N\{1}.In the above case, we say that A is non-trivial if it contains at least two points.

Then the following was proved in [65]:

Proposition 3.6. Let (X,T ) be a TDS with A a non-trivial weakly mixing subset of order 2. Then

A is also weakly mixing of order 2 and A is a compact space without isolated points and contains an

uncountable scrambled subset.


When considering the one-dimensional case, we have the following nice characterization (see [64]). By a

topological graph we mean a compact connected metric space homeomorphic to a polyhedron (a geometric

realization) of some finite one dimensional complex.

Theorem 3.7. Let (X,T ) be a TDS acting on a topological graph X. Then (X,T ) has positive

topological entropy if and only if (X,T ) contains a non-trivial weakly mixing subset of order 2.

Note that for a TDS acting on a topological graph, the system has positive topological entropy if and

only if it contains an intrinsic topological sequence entropy tuple with length 3 (see [72, 73]).

Moreover, (topological) weak mixing of subsets reflects a strong version of Li-Yorke chaos (see [65]).

Strictly speaking, the existence of a non-trivial weakly mixing subset implies uniform chaos (which is

an extended version of chaos in the sense of Li and Yorke, introduced recently in [3]). For some related

results see [3].

It turns out that the existence of a non-trivial weakly mixing subset follows from relative (measure-

theoretic) weak mixing, meanwhile, the relative (measure-theoretic) weak mixing follows from positive

relative (measure-theoretic) entropy, as interpreted by the following result, which was proved in [64,65,78].

Observe that the first item of Theorem 3.8 is well known, see for example [26]. For some variations of

Theorem 3.8 in the absolute case, see [36].

Theorem 3.8. Let (X,T ) be a TDS, A a T -invariant sub-σ-algebra of BX and μ ∈ Me(X,T ). And

then let π : (X,BX , μ, T ) → (Z,Z, η, R) be the Pinsker factor of (X,BX , μ, T ) with respect to A with

μ =∫Zμzdη(z) the disintegration of μ over (Z,Z, η, R) (see [26] for the existence of such a disintegration).

Assume hμ(T,X | A) > 0. Then

(1) π : (X,BX , μ, T ) → (Z,Z, η, R) is a weakly mixing extension.

(2) supp(μz) is a non-trivial weakly mixing subset (in particular, it is a compact space without isolated

points) for η-almost every z ∈ Z.

(3) For η-almost every z ∈ Z, there exists an uncountable scrambled subset Kz ⊆ supp(μz) which is

dense in supp(μz).

(4) hμ(T,X | A) � infE⊆Z,η(E)=0 supz∈Z\E htop(T, supp(μz)).

Observe that Theorem 3.7 will not be valid in the general case. In fact, there exists a minimal TDS

such that it contains a non-trivial weakly mixing set of order 2 and it does not contain any non-trivial

weakly mixing set of order 3 (see [64, Example 1.4]). In particular, as a direct corollary of Theorem

3.8, with the help of the classical variational principle concerning entropy, the constructed TDS has zero

topological entropy. This result was generalized to arbitrary n in [65]: for each n ∈ N\{1}, there exists

a TDS containing a weakly mixing subset of order n but not order n+ 1 (see [65, Example 1.1]).

As a direct corollary of Proposition 3.6 and Theorem 3.8, one hasthe following corollary:

Corollary 3.9. Let π : (X,T ) → (Y, S) be a factor map between TDSs. Then

(1) htop(T,X | π) = suphtop(T,K), where the supremum is taken over all subsets K which is a

scrambled subset contained in some fiber of π.

(2) Assume htop(T,X | π) > 0. Then, for some y ∈ Y , there exists an uncountable scrambled subset

contained in π−1(y).

Proof. For each μ ∈ Me(X,T ), let π1 : (X,BX , μ, T ) → (Z,Z, η, R) be the Pinsker factor of (X,BX , μ,

T ) with respect to π−1BY with μ =∫Z μzdη(z) the disintegration of μ over (Z,Z, η, R). Then there exists

a homomorphism π2 : (Z,Z, η, R) → (Y,BY , ν, S) with ν = πμ satisfying π = π2 ◦ π1. Then supp(μz)

is contained in some fiber of π for η-almost every z ∈ Z. Now the conclusion follows from the relative

variational principle, Proposition 3.6 and Theorem 3.8.

In particular, by letting the factor map be trivial, Corollary 3.9 tells us that there is a topologically

‘rather big’ scrambled set in any TDS with positive topological entropy, which strengthens the result of

the existence of an uncountable scrambled set.


In [41] Huang and Ye also proved that any non-trivial 2-scattering TDS is Li-Yorke chaotic. With the

help of the well-known Mycielski Theorem (see [62]), this result can also be generalized to the relative

case by introducing the relative 2-scattering.

Definition 3.10. Let π : (X,T ) → (Y, S) be a factor map between TDSs. π is called relative 2-

scattering if

limn→+∞N

( n−1∨

i=0

T−i{U1c, U2

c} | π)

= +∞

whenever U1, U2 are two non-empty open subsets of X with disjoint closures satisfying π(U1)∩π(U2) �= ∅.A set K of a compact metric space X is a Mycielski set if it has the form of K =

⋃+∞j=1 Cj with each

Cj a Cantor set of X . The following is a variation of the Mycielski Theorem.

Proposition 3.11. Let X be a compact metric space without isolated points. If R ⊆ X2 contains a

dense Gδ subset of X2, then there exists a dense Mycielski subset K ⊆ X satisfying K2\Δ2(K) ⊆ R.

Before proceeding, let us introduce some definitions.

Definition 3.12. Let (X,T ) be a TDS. A pair (x1, x2) ∈ X2 is called asymptotic (denoted by (x1, x2) ∈AR(X,T )) if limn→+∞ d(T nx1, T

nx2) = 0.

Definition 3.13. Let π : (X,T ) → (Y, S) be a factor map between TDSs.

(1) π is called relatively sensitive if there exists ε > 0 such that for each δ > 0 and x ∈ X , there exists

(x1, x2) ∈ R(2)π with d(x, x1) < δ, d(x, x2) < δ and d(T nx1, T

nx2) > ε for some n ∈ Z+.

(2) (X,T ) is sensitive if there exists ε > 0 such that for each δ > 0 and x ∈ X , there exists x′ ∈ X

with d(x, x′) < δ and d(T nx, T nx′) > ε for some n ∈ Z+.

Obviously, if π is relatively sensitive then (X,T ) is sensitive.

Then it was proved in [79] that under some necessary assumptions relative 2-scattering implies Li-

Yorke chaos on fibers, which generalizes the fact that each non-trivial 2-scattering TDS is Li-Yorke

chaotic (observe that each 2-scattering TDS is transitive).

Recall that the map f : X → Y between topological spaces is open if f(U) is an open subset of Y

whenever U is a non-empty open subset of X .

Theorem 3.14. Let π : (X,T ) → (Y, S) be an open factor map between TDSs, where π has relative

2-scattering. Suppose that there exists a second category subset Y0 ⊆ Y such that, for each y ∈ Y , there

exists a non-empty open subset Ly of π−1(y) satisfying that Ly contains no isolated points. Then there

exists an uncountable scrambled subset in some fiber of π, once one of the following properties holds:

(1) (AR(X,T ) ∩R(2)π )\Δ2(X) is a first category subset in R

(2)π .

(2) π is relatively sensitive.

(3) (X,T ) is minimal.

(4) (X,T ) is a transitive TDS which is not sensitive.

4 Asymptotic pairs in fibers over a given factor

Classically in topological dynamics, one considers the asymptotic behavior of pairs of points. For any

TDS with infinitely many points, it is not hard to prove the existence of a proper asymptotic pair in

a symbolic system (recall that an asymptotic pair is proper if it consists of different points), which was

extended to expansive homeomorphisms [16] by Bryant in 1960 (recall that a TDS (X,T ) is expansive if

there exists δ > 0 such that, for each (x1, x2) ∈ X2\Δ2(X), d(T nx1, Tnx2) > δ for some n ∈ Z).

Whereas, in [41] it was proved that the set of all asymptotic pairs of any transitive infinite TDS is

‘small’ in the sense of being of first category.

Theorem 4.1. Let (X,T ) be a transitive infinite TDS. Then AR(X,T ) ⊆ X2 is of first category, and

additionally, {y ∈ X : (x, y) ∈ AR(X,T )} ⊆ X is also of first category for each x ∈ X.


And then in [11] Blanchard, Host and Ruette proved that in any TDS with positive topological entropy,

there is a measure-theoretically ‘rather big’ set of proper asymptotic pairs. In fact, using ergodic methods

and by constructing an excellent partition, they showed that if μ is a T -ergodic invariant measure with

positive measure-theoretic entropy, then μ-almost every point belongs to some proper asymptotic pair,

which implies the existence of a proper asymptotic pair in any TDS with positive topological entropy

from the classical variational principle concerning entropy. In fact, the result proved in [11] is much more

than this. Anosov diffeomorphisms T on a manifold have stable and unstable foliations: points belonging

to the same stable foliation are asymptotic under T and tend to diverge under T−1, while pairs belonging

to the unstable foliation behave in the opposite way. Blanchard, Host and Ruette showed that any TDS

with positive topological entropy retains a faint flavour of this situation.

Theorem 4.2. Let (X,T ) be a TDS with positive topological entropy. Then there exist δ > 0, an

uncountable subset F ⊆ X and an uncountable subset Fx ⊆ X for each x ∈ F such that if x ∈ F and

y ∈ Fx then

limn→+∞ d(T nx, T ny) = 0, lim inf

n→+∞ d(T−nx, T−ny) = 0 and lim supn→+∞

d(T−nx, T−ny) � δ.

Along similar ideas, these results can be obtained in the relative setting (see [78]). That is, for any given

factor map between TDSs, if the factor map has positive relative topological entropy, then there exists

a proper asymptotic pair in some fiber of it. In fact, combined with the relative variational principle, it

follows from the theorem below.

Theorem 4.3. Let π : (X,T ) → (Y, S) be a factor map between TDSs and μ ∈ Me(X,T ) satisfy

hμ(T,X | π) > 0. Then for μ-almost every x ∈ X, there exists y ∈ X\{x} with (x, y) ∈ AR(X,T )∩R(2)π .

Given a factor map between TDSs, a basic fact states that if topological entropy of the extension is

bigger strictly than that of the factor, then relative topological entropy of the factor map is positive.

Thus, as a direct corollary, we have the following:

Corollary 4.4. Let π : (X,T ) → (Y, S) be a factor map between TDSs. If htop(T,X) > htop(S, Y ),

then

(AR(X,T ) ∩R(2)π )\Δ2(X) �= ∅.

In the process of proving Theorem 4.3, as shown in Proposition 4.5, the concept of relative Pinsker

σ-algebra plays a key role.

Let (X,B, μ, T ) be an MDS and ξ a measurable partition of it. Recall that ξ is generating if ξT is equal

to B outside a set of measure zero.

Proposition 4.5. Let (X,T ) be a TDS, μ ∈ M(X,T ) and A a T -invariant sub-σ-algebra. Then

(X,BX , μ, T ) admits a generating partition P with P ⊇ A such that any pair of points belonging to the

same atom of P− is asymptotic and+∞⋂

n=0

T−nP− = Pμ(A).

Additionally, if hμ(T,X | A) > 0 then the σ-algebras P− and BX do not coincide up to sets of μ-measure

zero.

Thus, in the notations of Proposition 4.5, the construction of the measurable partition P will be crucial,

and it is obtained by setting P = Q ∨W , where both Q and W are measurable partitions, the partition

Q is constructed by some standard techniques in ergodic theory satisfying

+∞⋂

n=0

(T−nQ− ∨A) ⊆ Pμ(A),

and the partition W is constructed satisfying that W (as a sub-σ-algebra of BX) coincides with A outside

a set of measure zero. Then Proposition 4.5 follows from the following fundamental fact about the relative

Pinsker σ-algebra.


Proposition 4.6. Let (X,B, μ, T ) be an MDS and A a T -invariant sub-σ-algebra. If P is a measurable

generating partition satisfying P ⊇ A, then

Pμ(A) ⊆+∞⋂

n=0

T−nP−.

In fact, we can strengthen these results with Proposition 4.8 (proved in [45]).

Before proceeding, let’s make some preparations. Let (X,B, μ, T ) be an MDS, A a T -invariant

sub-σ-algebra of B and n ∈ N\{1}. We define a new invariant measure λAn (μ) on the product space

(Xn,Bn, T (n)) determined completely by (A1, . . . , An ∈ B)

λAn (μ)

( n∏

i=1

Ai

)

=

∫

X

n∏

i=1

E(1Ai | Pμ(A))(x)dμ(x).

In fact, let π : (X,B, μ, T ) → (Z,Z, η, R) be the Pinsker factor of (X,B, μ, T ) with respect to A with

μ =∫Zμzdη(z) the disintegration of μ over (Z,Z, η, R). Then

λAn (μ) =

∫

Z

μz × · · · × μzdη(z) (n-times).

In the notations as above, one standard but very useful fact is the following (see also [43] for it in the

absolute case):

Proposition 4.7. Let (X,B, μ, T ) be an MDS, A a T -invariant sub-σ-algebra and α = {A1, . . . , An} ∈PX , n ∈ N\{1}. Then hμ(T, α | A) > 0 if and only if

λAn (μ)

( n∏

i=1

Aci

)

> 0.

With the help of this, we can prove (see also [11] for the absolute case):

Theorem 4.8. Let π : (X,T ) → (Y, S) be a factor map between TDSs and α = {A1, A2} ∈ PX . If

hμ(T, α | π) > 0 for some μ ∈ M(X,T ), then

A1 ×A2 ∩ AR(X,T ) ∩R(2)π �= ∅.

As a corollary, by standard arguments we obtained directly:

Corollary 4.9. Let π2 : (X,T ) → (Y, S) and π1 : (Y, S) → (Z,R) be two factor maps between TDSs.

If

R(2)π2

⊇ AR(X,T ) ∩R(2)π1◦π2 ,

then htop(S, Y | π1) = 0.

In particular, letting (Y, S) = (X,T ) and π2 be the identity map, one has the following corollary:

Corollary 4.10. Let π : (X,T ) → (Y, S) be a factor map between TDSs. If each pair from AR(X,T )∩R

(2)π is trivial (i.e., not proper), then htop(T,X | π) = 0.

Even so, it stands as a challenge if there exists a purely topological proof of the existence of a proper

asymptotic pair in any TDS with positive topological entropy.

Let (X,T ) be a TDS. Each pair of AR(X,T ) is also called a positively asymptotic pair, and each pair

from AR(X,T ) ∩AR(X,T−1) is called a doubly asymptotic pair.

Another long standing question is whether there exists a proper doubly asymptotic pair in any TDS

with positive topological entropy under some necessary assumptions. I should note that Wen Huang [37]

told me that he constructed a TDS with positive topological entropy such that each doubly asymptotic

pair of it is trivial (i.e., not proper). See also [17, 59] for some related results for algebraic actions of a


general amenable group. However, the TDS he constructed is not expansive. So we conjecture that each

infinite expansive TDS with positive topological entropy (or some special subclass of it) contains proper

doubly asymptotic pairs.

Part IILocal theory of relative entropy

5 Local variational principles concerning relative entropy

In the establishment of variational relationships between two kinds of entropy pairs and tuples, a key

point is the following so-called local variational principles.

Let (X,T ) be a TDS and U ∈ CX . In [8] the authors showed that there is μ ∈ M(X,T ) satisfying

inf hμ(T, α) � htop(T,U), where the infimum is taken over all partitions α which are finer than U .And then the converse was proved in [32]. In fact in [43] the authors proved that for μ ∈ M(X,T ) if

hμ(T, α) > 0 for each partition α finer than U , then

infα∈PX ,αU

hμ(T, α) > 0

and htop(T,U) > 0, providing another kind of converse statement of the above mentioned result in [8].

To study the relationship between htop(T,U) and infα∈PX ,αU hμ(T, α) for each μ ∈ M(X,T ), in [69]

Romagnoli introduced two kinds of measure-thereotic entropy for finite Borel measurable covers, namely,

for U ∈ CBX and μ ∈ M(X,T ), where CB

X denotes the set of all finite Borel measurable covers of X ,

h+μ (T,U) = inf

α∈PX ,αUhμ(T, α)

and

h−μ (T,U) = lim

n→+∞1

ninf

α∈PX ,α∨n−1i=0 T−iU

Hμ(α)

(

= infn�1

1

ninf


Hμ(α)

)

,

where the limit as above exists and the second identity follows from the sub-additivity of the sequence

infα∈PX ,α∨n−1i=0 T−iU Hμ(α). By standard arguments, it is not hard to obtain that

h−μ (T,U) � h+

μ (T,U) � hμ(T,X) for U ∈ CBX

and

h−μ (T,U) � htop(T,U) for U ∈ CX .

Stronger relations between them were obtained by [69] and [32], respectively.

Theorem 5.1. Let (X,T ) be a TDS and U ∈ CX . Then

maxμ∈M(X,T )

h±μ (T,U) = max

μ∈Me(X,T )h±μ (T,U) = htop(T,U).

Observe that in [39] it was proved that if there exist a TDS (X,T ) and μ ∈ M(X,T ), U ∈ CX satisfying

h−μ (T,U) < h+

μ (T,U), then there exists a uniquely ergodic TDS with the same property (recall that TDS

(X,T ) is uniquely ergodic if M(X,T ) is just a singleton). Thus, combined with Theorem 5.1, one has

the following result which was observed in [44] for the first time in the literature.

Theorem 5.2. Let (X,T ) be a TDS and U ∈ CBX . Then h−

μ (T,U) = h+μ (T,U) for each μ ∈ M(X,T ).


Recall that the classical variational principle follows from the local ones by some simple arguments.

Also note that in a recent work (see [70]), Shapira presented an effective way of computing h+μ (T,U) and

then provided elegant new proofs for most of the results mentioned above.

For each U ∈ CBX we can also introduce htop(T,U) and show that

h−μ (T,U) = h+

μ (T,U) � htop(T,U), ∀μ ∈ M(X,T ).

The following result from [44] tells us that we could not expect to obtain a result similar to Theorem 5.1

for a general U ∈ CBX , as there are many transitive non-periodic TDSs with zero topological entropy.

Proposition 5.3. Let (X,T ) be a transitive non-periodic TDS. Then for any 1 > ε > 0 and each

N ∈ N\{1}, there exists a closed cover U of X consisting of N elements with htop(T,U) � (1− ε) logN .

For some related results see [20, 46, 49, 81].

When considering the relative case of a given factor map between TDSs, a natural question is whether

there are local relative variational principles, which was addressed and answered affirmatively in [44].

Let π : (X,T ) → (Y, S) be a factor map between TDSs and μ ∈ M(X,T ), U ∈ CBX . Following the

ideas from [69], similarly we can introduce

h+μ (T,U | π) = inf

α∈PX ,αUhμ(T, α | π)

and

h−μ (T,U | π) = lim

n→+∞1

ninf


Hμ(α | π−1BY ).

The following results tell us that the introduced entropy has good properties and can be used to

estimate the relative measure-theoretic entropy of systems (see [44]).

Proposition 5.4. Let π : (X,T ) → (Y, S) be a factor map between TDSs and μ ∈ M(X,T ). Then

hμ(T,X | π) = supU∈CX

h−μ (T,U | π).

Proposition 5.5. Let π : (X,T ) → (Y, S) be a factor map between TDSs, μ ∈ M(X,T ) and U ∈ CBX .

If μ =∫Ω μωdη(ω) is the ergodic decomposition of μ, then

h±μ (T,U | π) =

∫

Ω

h±μω

(T,U | π)dη(ω).

In particular, the functions

h±• (T,U | π) : M(X,T ) → [0, htop(T,U | π)]

are both affine.

Then Theorem 5.1 was generalized to the relative case as follows.

Theorem 5.6. Let π : (X,T ) → (Y, S) be a factor map between TDSs and U ∈ CX . Then htop(T,U |π) = h−

μ (T,U | π) for some μ ∈ M(X,T ). Moreover,

htop(T,U | π) = maxμ∈M(X,T )

h−μ (T,U | π) = max

μ∈Me(X,T )h−μ (T,U | π).

In fact, following the ideas from [32], we can obtain a much stronger result. Remark that Theorem 5.7

has been proved by [32] in the absolute case.

Theorem 5.7. Let π : (X,T ) → (Y, S) be a factor map between TDSs and U ∈ CX . Then h+μ (T,U |

π) � htop(T,U | π) for each μ ∈ M(X,T ) and

infα∈PX ,αU

supμ∈M(X,T )

hμ(T, α | π) = htop(T,U | π).


Combined with Proposition 5.4, the relative variational principle follows from the local relative varia-

tional principles by some standard arguments.

As it seems that the methods in [8, 69] (especially the combinatorial lemma in [8]) are difficult to

generalize to the relative case, we should find some new methods quite different from the ones used there,

which were influenced by [23,61]. As the substitute of the combinatorial lemma in [8], the following selec-

tion lemma, appearing first in [44], plays a key role in the construction of an invariant Borel probability

measure with the required properties, which is also crucial in [20, 46, 49, 81].

Lemma 5.8 (The Selection Lemma). Let π : (X,T ) → (Y, S) be a factor map between TDSs and

y ∈ Y, U ∈ CX . Assume that α1, . . . , αK , K ∈ N is a finite sequence in PX finer than U . Then for each

n ∈ N, there exists a finite subset Bn ⊆ π−1(y) with cardinality

[1

KN

( n−1∨

i=0

T−iU , π−1(y)

)]

such that each atom of∨n−1

i=0 T−iαl contains at most one point from Bn for all 1 � l � K, where [•]denotes the integer part of •.

Additionally, with the help of the relative version of results from [32,39], by the relative Jewett-Krieger

Theorem [75], we obtain the following:

Proposition 5.9. Let π : (X,T ) → (Y, S) be a factor map between TDSs and μ ∈ M(X,T ), U ∈ CBX .

Then

h−μ (T,U | π) = h+

μ (T,U | π)(from now on, denoted by hμ(T,U | π) and called the relative measure-theoretic μ-entropy of U relevant

to π).

In fact, following similar ideas we could prove an inner version of Theorem 5.6.

Let π : (X,T ) → (Y, S) be a factor map between TDSs and U ∈ CX . Set

h(T,U | y) = lim supn→+∞

1

nlogN

( n−1∨

i=0

T−iU , π−1y

)

and h(T,U | π) = supy∈Y h(T,U | y). We have directly that

htop(T, π−1y) = suph(T,V | y),

where the supremum is taken over all V ∈ CX . Moreover, it is easy to check that the map y �→logN(U , π−1y) is Borel measurable and for each n,m ∈ N and y ∈ Y , one has

logN

( n+m−1∨

i=0

T−iU , π−1y

)

� logN

( n−1∨

i=0

T−iU , π−1y

)

+ logN

(m−1∨

i=0

T−iU , π−1(Sny)

)

.

Furthermore, by Kingman’s sub-addtive ergodic theorem (see [74]), one has that for any ν ∈ M(Y, S),

the limit of the sequence{1

nlogN

( n−1∨

i=0

T−iU , π−1y

)

: n ∈ N

}

exists and is equal to h(T,U | y) for ν-almost every y ∈ Y .

The inner version of local relative variational principle Theorem 5.6 is stated as follows (for details the

reader is referred to [80], for some related topic see [60]).


Theorem 5.10. Let π : (X,T ) → (Y, S) be a factor map between TDSs and U ∈ CX , η ∈ M(Y, S).

Then ∫

Y

h(T,U | y)dη(y) = maxξ∈M(X,T ),πξ=η

hξ(T,U | π)

and

htop(T,U | π) = h(T,U | π) = supν∈M(Y,S)

∫

Y

h(T,U | y)dν(y)

= supν∈Me(Y,S)

∫

Y

h(T,U | y)dν(y).

We end this section with a stronger version of Proposition 4.7 (see [45]) (see also [43] for it in the

absolute case).

Proposition 5.11. Let π : (X,T ) → (Y, S) be a factor map between TDSs and μ ∈ M(X,T ), U =

{U1, . . . , Un} ∈ CBX , where n ∈ N\{1}. Set A = π−1BY . Then the following items are equivalent:

(1) hμ(T,U | π) > 0.

(2) λAn (μ)(

∏ni=1 U

ci ) > 0.

(3) hμ(T, α | π) > 0 whenever α ∈ PX is finer than U .

6 Relative entropy tuples

At the beginning of the 1990s, Blanchard began his study of the counterpart of a Kolmogorov system

in topological dynamics. First, he tried to define topological Kolmogorov systems by means of global

notions as follows (see [6]).

Let (X,T ) be a TDS and U ∈ CX . U is called non-trivial if the closure of each element of the cover is

not the whole space X .

Definition 6.1. Let (X,T ) be a TDS. We say that (X,T )

(1) has UPE if htop(T,U) > 0 for each non-trivial U ∈ CX consisting of two elements.

(2) has CPE if each non-trivial factor has positive topological entropy.

The notion of disjointness of two systems was introduced by Furstenberg in [25]. In ergodic theory,

a basic fact states that each Kolmogorov system is disjoint from all MDSs with zero measure-theoretic

entropy. In order to better understand how topological entropy is woven into the general pattern of

topological dynamics and test whether the notions of UPE and CPE favour the interplay between entropy

and other properties from topological dynamics, Blanchard realized that a local viewpoint may be very

useful (see [7]). Let’s recall the following definition from [7].

Definition 6.2. Let (X,T ) be a TDS and (x1, x2) ∈ X2. (x1, x2) is called an entropy pair if x1 �= x2

and htop(T, {U c1 , U

c2}) > 0 whenever U1 and U2 are closed neighborhoods of x1 and x2, respectively, with

U1 ∩ U2 = ∅.Then the global positivity of topological entropy is localized as follows (see [7]).

Proposition 6.3. Let (X,T ) be a TDS. Then (X,T ) has positive topological entropy if and only if it

contains an entropy pair.

By localizing the positivity of topological entropy as above, he could prove the counterpart of disjoint-

ness result in topological dynamics (see [7]). From then on, more and more people came into the study

of the so called local entropy theory of dynamical systems: firstly entropy pairs for an invariant measure

and its variational relationship with topological entropy pairs (see [8, 10, 27]), second entropy tuples in

both settings and the variational relationship between them (see [43]), then entropy sets [12,22] and last

entropy points [12,77] (and the references therein). As all of these results were generalized to the relative


case [44,45] which will be discussed later, we elect to skip them temporarily and return to some of them

in detail later, save the following results.

Definition 6.4. Let (X,T ) be a TDS and ∅ �= K ⊆ X . We say that K is an entropy set with respect to

(X,T ) if K is non-trivial (i.e., not a singleton) and htop(T,U) > 0 whenever U ∈ CX satisfying K\U �= ∅for each U ∈ U .

In [43] the authors obtained a characterization of topological entropy tuples in terms of the following

weak Bernoulli property.

Theorem 6.5. Let (X,T ) be a TDS and (x1, . . . , xn) ∈ Xn\Δn(X), where n ∈ N\{1}. Then

(x1, . . . , xn) is an entropy n-tuple if and only if whenever Ui is a neighborhood of xi, i = 1, . . . , n, there

exists D = {d1 < d2 < · · · } ⊆ N with positive density such that⋂+∞

i=1 T−diUs(i) �= ∅ once s(j) ∈ {1, . . . , n}for each j ∈ N.

With the help of a simple characterization of measure-theoretic entropy tuples (inspired by [27]) in [22],

the authors proved the following result that there exists an uncountable entropy set in any TDS with

positive topological entropy.

Proposition 6.6. Under the same assumptions as in Theorem 3.8 and letting A = {∅, X}, we have

that supp(μz) is an entropy set for η-almost every z ∈ Z.

Even so, there exists a transitive TDS with positive topological entropy for which there is a maximal

entropy set consisting of exactly two points (see [22]).

Observe that by the classical variational principle, each TDS consisting of at most countably many

points must have zero topological entropy. In the study of entropy points in [77], the authors proved the

following result.

Theorem 6.7. Let (X,T ) be a TDS. Then there is a countable closed subset such that its entropy is

equal to the topological entropy of the whole system. Moreover, the constructed subset can be chosen so

that its limit set has at most one limit point.

In fact, starting from this result, in [47,48] the authors presented a new dynamical characterization of

asymptotically h-expansive TDSs.

In the following, we will see how the above mentioned results were generalized to the relative case of

a given factor map between TDSs, for details see [45].

The first attempt along these lines was due to Glasner and Weiss [31]. Let π : (X,T ) → (Y, S) be

a factor map between TDSs and denote by E2(X,T ) the set of all topological entropy pairs of (X,T ).

In order to introduce the notion of relative topological Pinsker factor, as a generalization of topological

Pinsker factor introduced by Blanchard and Lacroix in [13] (we shall turn to it in more details in next

section), in [31] the authors defined the relative topological entropy pairs by E2(X,T )∩R(2)π and showed

that

htop(Tπ, Xπ) = htop(S, Y ),

where (Xπ, Tπ) is the factor of (X,T ) induced by the invariant closed equivalence relation (ICER) gen-

erated by

(E2(X,T ) ∩R(2)π ) ∪Δ2(X).

That is, (Xπ, Tπ) just shows the place in the system of factors between (X,T ) and (Y, S) where E2(X,T )

begins to appear on the fibers, alternatively, (Xπ, Tπ) is the ‘greatest’ topological factor between (X,T )

and (Y, S) whose fibers contain no elements from E2(X,T ). However, the shortcoming of this notion

is that it comes from topological entropy not from relative (topological) entropy; moreover, as for the

notions of UPE and CPE extensions based on this [31], unlike in the absolute case a UPE extension need

not be a CPE extension even if we only consider minimal TDSs, for details see [31].

Thus, along a different line of considering relative topological and measure-theoretic entropy from

that of Glasner and Weiss, in the following we will introduce the notions of relative topological entropy


pairs and tuples in both settings and build the variational relationship between them with the help of

the local variational principles concerning relative entropy. See [45] for details of the whole story, see

also [7, 10, 27, 43] for them in the absolute case.

Definition 6.8. Let π : (X,T ) → (Y, S) be a factor map between TDSs, μ ∈ M(X,T ) and (x1, . . . , xn)

∈ Xn\Δn(X), where n ∈ N\{1}. (x1, . . . , xn) is called

(1) a relative topological entropy n-tuple relevant to π if htop(T,U | π) > 0 whenever U ∈ CX satisfies

{x1, . . . , xn}\U �= ∅ for each U ∈ U .(2) a relative measure-theoretical entropy n-tuple for μ relevant to π if hμ(T,U | π) > 0 whenever

U ∈ CX satisfies {x1, . . . , xn}\U �= ∅ for each U ∈ U .Denote by En(X,T | π) and Eμ

n(X,T | π) the set of all relative topological entropy n-tuples and

measure-theoretical entropy n-tuples for μ relevant to π, respectively. When π is trivial, we omit the

restriction of π recovering those notions in the absolute case.

Thus the positivity of relative topological entropy can be localized and characterized by the notion of

relative topological entropy tuples as follows.

Proposition 6.9. Let π : (X,T ) → (Y, S) be a factor map between TDSs and n ∈ N\{1}. Assume

htop(T,X | π) > 0. Then En(X,T | π) ⊆ R(n)π is a non-empty invariant subset and

En(X,T | π)\Δn(X) = En(X,T | π).In fact, if U = {U1, . . . , Un} ∈ CX satisfies htop(T,U | π) > 0, then

En(X,T | π) ∩n∏

i=1

U ci �= ∅.

The following lift-up and projection properties of relative entropy tuples are not hard to obtain, which

is the base of the notion of (relative) topological Pinsker factor introduced and discussed in next section.

Proposition 6.10. Let π1 : (X,T ) → (Y, S) and π2 : (Y, S) → (Z,R) be factor maps between TDSs

and n ∈ N\{1}, μ ∈ M(X,T ), ν.= π1μ ∈ M(Y, S). Then

(1)

Eνn(Y, S | π2) ⊆ (π1 × · · · × π1)E

μn(X,T | π2 ◦ π1) ⊆ Eν

n(Y, S | π2) ∪Δn(Y )

and

En(Y, S | π2) ⊆ (π1 × · · · × π1)En(X,T | π2 ◦ π1) ⊆ En(Y, S | π2) ∪Δn(Y ).

(2)

Eμn(X,T | π1) ⊆ Eμ

n(X,T | π2 ◦ π1)

and

En(X,T | π1) ⊆ En(X,T | π2 ◦ π1).

As a direct corollary of Proposition 5.11, we have the following useful observation.

Corollary 6.11. Let π : (X,T ) → (Y, S) be a factor map between TDSs and μ ∈ M(X,T ), n ∈ N\{1}.Set λπ

n(μ) = λπ−1BYn (μ). Then

Eμn(X,T | π) = supp(λπ

n(μ))\Δn(X) ⊆ R(n)π .

Remark that Corollary 6.11 in the absolute case was first proved by Glasner in [27] for n = 2 and then

by Huang and Ye in [43] for a general n ∈ N\{1}.With the help of the local variational principles concerning relative entropy built in the previous section,

we have the following variational relationship.

Theorem 6.12. Let π : (X,T ) → (Y, S) be a factor map between TDSs. Then

(1) Eμn(X,T | π) ⊆ En(X,T | π) for any μ ∈ M(X,T ) and each n ∈ N\{1}.

(2) there exists μ ∈ M(X,T ) such that En(X,T ) = Eμn(X,T | π) for each n ∈ N\{1}.


As a direct corollary of Corollary 6.11 and Theorem 6.12, we have the following corollary:

Corollary 6.13. Let π : (X,T ) → (Y, S) be a factor map between TDSs. If htop(T,X | π) > 0, then

there exists (x1, . . . , xn) ∈ En(X,T | π) such that xi �= xj if i �= j.

As in the absolute case, we can also introduce the notion of relative entropy set.

Definition 6.14. Let π : (X,T ) → (Y, S) be a factor map between TDSs and ∅ �= K ⊆ X . We say

that K is a relative entropy set relevant to π if K is non-trivial and htop(T,U | π) > 0 whenever U ∈ CXsatisfying K\U �= ∅ for each U ∈ U .

Then we have a result similar to Proposition 6.6.

Proposition 6.15. Let φ : (X,T ) → (Y, S) be a factor map between TDSs and set A = φ−1BY . Then

under the same assumptions as in Theorem 3.8 we have that supp(μz) is a relative entropy set relevant

to φ for η-almost every z ∈ Z.

With the help of Theorem 6.12, we can relate the relative topological entropy pairs to asymptotic pairs

on fibers as follows.

Proposition 6.16. Let π : (X,T ) → (Y, S) be a factor map between TDSs. Then

E2(X,T | π) ⊆ AR(X,T ) ∩R(2)π .

7 Relative topological Pinsker factor, relative UPE and CPE extensions

Although as mentioned in Subsection 2.4, in ergodic theory the four notions of a Kolmogorov system are

equivalent, in the topological setup of Definition 6.1, we only have the implication that UPE implies CPE.

Examples of CPE but not UPE TDSs were given by Blanchard in [6], and then recently such an example

in the minimal case was given in [71] (for the analogous relative question see [31]). Moreover, by exploring

finer structures of UPE in [43], i.e., UPE of order n for each n ∈ N\{1}, the authors provided a TDS which

is UPE of order 2 but is not UPE of order 3 (see [43]). For related results see [6,7,13,31,43,45,51,56,67,71].

All of this tells us that a more systematic study of these UPE, CPE TDSs and other related dynamical

properties (in the relative setting) is deserved, which was done in [45]. Thus, in this section we aim to

overview those results, which are applications of the theory built in previous sections, including relative

topological Pinsker factor, relative UPE and CPE extensions.

7.1 Relative topological Pinsker factor

In ergodic theory, given an MDS, it is well known that a maximal factor with zero measure-theoretic

entropy exists.

Recall that by an ICER (introduced in the previous section) we mean an invariant closed equivalence

relation. By localizing topological entropy in topological dynamics, Blanchard introduced the notion of

topological entropy pairs (see [7]), and then based on this, Blanchard and Lacroix showed the existence

of a largest factor with zero topological entropy for any given TDS (called the topological Pinsker factor)

(see [13]).

Theorem 7.1. Let (X,T ) be a TDS and set (XP , TP ) as the TDS induced by the ICER generated

by E2(X,T ) ∪Δ2(X). Then (XP , TP ) is the maximal topological factor of (X,T ) with zero topological

entropy in the sense that

(1) (XP , TP ) is a factor of (X,T ) (let πP : (X,T ) → (XP , TP ) be the factor map between them) with

zero topological entropy; and

(2) if π : (X,T ) → (Y, S) is a factor map between TDSs with htop(S, Y ) = 0, then there exists a factor

map φ : (XP , TP ) → (Y, S) satisfying π = φ ◦ πP .


Let π : (X,T ) → (Y, S) be a factor map between TDSs. Recall from the previous section, that in [31]

Glasner and Weiss made the first attempt on the notion of the relative topological Pinsker factor (called

relative topological Pinsker1 factor) and obtained a TDS (Xπ, Tπ) by collapsing E2(X,T ) ∩ R(2)π , all

topological entropy pairs on fibers, i.e., (Xπ, Tπ) is the ‘largest’ topological factor between (X,T ) and

(Y, S) whose fibers contain no elements from E2(X,T ), and it was shown that htop(Tπ, Xπ) = htop(S, Y ).

Later on in [56] Lemanczyk and Siemaszko presented another approach leading to the definition of the

relative topological Pinsker factor (called relative topological Pinsker2 factor temporarily). Let P ⊇Δ2(X) be the smallest ICER contained in R

(2)π satisfying that

hμ(TPπ , XP

π ) = hπP (μ)(S, Y ), ∀μ ∈ M(XPπ , TP

π ),

where (XPπ , TP

π ) is the TDS induced by P and πP : (XPπ , TP

π ) → (Y, S) denotes the natural homo-

morphism (note P ⊆ R(2)π ). Just after [56], in [67] Park and Siemaszko interpreted the relative topo-

logical Pinsker2 factor using relative measure-theoretic entropy and proved that P was generated by⋃

μ∈M(X,T ) supp(λπ2 (μ)). Thus by Theorem 6.12, one has (see [45]):

Proposition 7.2. Let π : (X,T ) → (Y, S) be a factor map between TDSs. Then (XPπ , TP

π ) is just the

TDS induced by the ICER generated by E2(X,T | π) ∪Δ2(X).

Observe that by Proposition 6.10, one has that

E2(X,T | π) ⊆ E2(X,T ) ∩R(2)π

and so the proposition holds:

Proposition 7.3. Let π : (X,T ) → (Y, S) be a factor map between TDSs. Then (Xπ , Tπ) is always a

factor of (XPπ , TP

π ).

That is, (XPπ , TP

π ) is the ‘greatest’ topological factor between (X,T ) and (Y, S) whose fibers contain

no relative topological entropy pair, which implies htop(TPπ , XP

π | πP ) = 0 by Proposition 6.10 and

consequently TDSs (XPπ , TP

π ) and (Y, S) have the same topological entropy. By Proposition 7.2, it seems

that the relative topological Pinsker2 factor is a more natural notion, as it is characterized by the relative

(topological) entropy. We will refer to it as the relative topological Pinsker factor.

7.2 Relative UPE and CPE extensions

It is well known in ergodic theory that a Kolmogorov system has a very strong mixing property.

Let (X,T ) be a TDS. Denote⋃

μ∈M(X,T ) supp(μ) by supp(X,T ), and note that there always exists

μ ∈ M(X,T ) with supp(X,T ) = supp(μ). Although, CPE does not imply any kind of mixing, not even

transitivity, it is shown that each CPE TDS is fully supported (see [6]).

Theorem 7.4. Let (X,T ) be a TDS. If (X,T ) has CPE, then supp(X,T ) = X.

In fact, CPE can be characterized by the following result (see [7]).

Proposition 7.5. Let (X,T ) be a TDS. Then (X,T ) has CPE if and only if X2 is the smallest ICER

containing E2(X,T ) ∪Δ2(X).

We should remark that in general we could not expect E2(X,T )∪Δ2(X) to be an equivalence relation

even if the considered (X,T ) is a minimal TDS, which was first explored in [31] (for details see [31,

Proposition 3.1]).

Compared to CPE, UPE seems to be a more successful counterpart of Kolmogorov systems in topolog-

ical dynamics. For example, aside from the strong mixing property, each Kolmogorov system is disjoint

from all MDSs with zero measure-theoretic entropy. For UPE we have similar results (see [6, 7]).

Let (X,T ) be a TDS. (X,T ) is called weakly mixing if (X2, T (2)) is a transitive TDS.

The following result was proved in [6].

Theorem 7.6. Any UPE TDS is weakly mixing.


Observe that by exploring finer structure of UPE TDSs, it was shown that there exists a topological

K-system which is not strongly mixing, whereas, each minimal topological K-system is strongly mixing

(see [42]).

The notion of disjointness of two TDSs was introduced in [25]. Let πX : (X,T ) → (Y, S) and πZ :

(Z,R) → (Y, S) be two factor maps between TDSs, and π1 : X × Z → X , π2 : X × Z → Z be the

projections. J ⊆ X×Z is called a joining of (X,T ) and (Z,R) over (Y, S) if J is a closed T ×R-invariant

subset satisfying

π1(J) = X, π2(J) = Z and π1 × π2(J) = Δ2(Y ).

Define that

X ×Y Z =⋃

y∈Y

π−1X (y)× π−1

Z (y).

Clearly, X ×Y Z is a joining of (X,T ) and (Z,R) over (Y, S). Call (X,T ) and (Z,R) disjoint over (Y, S)

if X ×Y Z contains no proper sub-joinings of (X,T ) and (Z,R) over (Y, S). When (Y, S) is trivial, we

also say that (X,T ) and (Z,R) are disjoint.

Then in [7] it was proved:

Theorem 7.7. Any UPE TDS is disjoint from every minimal TDS with zero topological entropy.

Again, all of these results can be generalized to the relative case as follows (see [45]).

Definition 7.8. Let π : (X,T ) → (Y, S) be a factor map between TDSs which is not a homeomorphism

and n ∈ N\{1}. We say that

(1) π has relative CPE if for any proper factor (Z,R) of (X,T ) with respect to (Y, S), it has positive

relative topological entropy with respect to (Y, S). In this case, we also say that (X,T ) has relative CPE

with respect to (Y, S).

(2) π has relative UPE of order n, if En(X,T | π) = R(n)π \Δn(X). When n = 2, we say simply that π

has relative UPE.

(3) π has relative topological K if it has relative UPE of order m for each m ∈ N\{1}.Just as in the absolute case, obviously relative UPE implies relative CPE, and both relative UPE and

CPE extensions are stable under factor maps.

Similar to Proposition 7.5, we have:

Proposition 7.9. Let π : (X,T ) → (Y, S) be a factor map between TDSs which is not a homeomor-

phism. Then π has relative CPE if and only if R(2)π is just the ICER generated by E2(X,T | π)∪Δ2(X).

Let π : (X,T ) → (Y, S) be a factor map between TDSs. Recall that π is weakly mixing if (R(2)π , T (2)) is

a transitive TDS, and weakly mixing of all orders if (R(n)π , T (n)) is a transitive TDS for each n ∈ N\{1}.

Then Theorem 7.4 can be generalized as follows.

Theorem 7.10. Let π : (X,T ) → (Y, S) be a factor map between TDSs which is not a homeomorphism.

Assume that π has relative CPE. Then

(1) Let μ ∈ M(X,T ). If Eμ2 (X,T | π) = E2(X,T | π), then supp(μ) = π−1(supp(πμ)) and π−1(y) is

a singleton for each y ∈ Y \supp(πμ).(2) π−1(y) is a singleton for each y ∈ Y \supp(Y, S).(3) π−1(supp(Y, S)) = supp(X,T ) and if π is weakly mixing, then supp(X,T ) = X.

With the help of the relative topological entropy pairs introduced, using Proposition 6.10, it is not

hard to obtain a relative disjointness result related to relative UPE.

Let π : (X,T ) → (Y, S) be a factor map between TDSs. We say that π is minimal if X is the only

closed T -invariant subset with π-image Y .

Theorem 7.11. Let πX : (X,T ) → (Y, S) and πZ : (Z,R) → (Y, S) be two factor maps between TDSs,

where πX is open but not invertible and πZ is minimal. Suppose that πX has relative topological K and

htop(R,Z|πZ) = 0. Then (X,T ) and (Z,R) are disjoint over (Y, S).


Let πi : (Xi, Ti) → (Y, S) (i = 1, 2) be two factor maps between TDSs. Then (X1 ×Y X2, T1 ×Y T2)

forms a TDS, where

T1 ×Y T2 : X1 ×Y X2 → X1 ×Y X2, (x1, x2) �→ (T1x1, T2x2).

Denote by π1 ×Y π2 the factor map induced naturally by π1 and π2, i.e.,

π1 ×Y π2 : (X1 ×Y X2, T1 ×Y T2) → (Y, S), (x1, x2) �→ π1(x1).

The above notations can be generalized to the case of any given n factor maps.

Theorem 7.12. Let πi : (Xi, Ti) → (Y, S) be a factor map between TDSs which is not a homeomor-

phism, i = 1, . . . , n, n ∈ N\{1}. Assume that each πi has relative UPE and supp(Y, S) = Y , i = 1, . . . , n.

Then π1×Y · · ·×Y πn also has relative UPE. If in addition all πi are open, then (X1×Y · · ·×Y Xn, T1×Y

· · · ×Y Tn) is a transitive TDS if and only if (Y, S) is a transitive TDS.

As a direct corollary, one has that relative UPE reflects some kind of relative mixing (in particular, it

is a strong version of Theorem 7.6).

Corollary 7.13. Let π : (X,T ) → (Y, S) be an open factor map between TDSs which is not a homeo-

morphism. If π has relative UPE, then the following are equivalent:

(1) π is weakly mixing.

(2) π is weakly mixing of all orders.

(3) (X,T ) is a transitive TDS satisfying supp(X,T ) = X.

(4) (Y, S) is a transitive TDS satisfying supp(Y, S) = Y .

Proof. The equivalence of (1), (2) and (4) is the second part of [45, Theorem 6.7]. The direction of

(3) ⇒ (4) is direct. Now we finish our proof by claiming the direction of (1) ⇒ (3). If π is weakly mixing,

then it is easy to check that (X,T ) is a transitive TDS. Moreover, observe that relative UPE implies

relative CPE and so by Theorem 7.10 (3), one has supp(X,T ) = X , thus we conclude (3).

Moreover, relative UPE and CPE are preserved under finite product (see [45]).

Proposition 7.14. Let πi : (Xi, Ti) → (Yi, Si) be a factor map between TDSs which is not a homeo-

morphism, i = 1, 2. Assume that both π1 and π2 have relative UPE (relative CPE, respectively). Then

the following statements are equivalent:

(1) π1 × π2 has relative UPE (relative CPE, respectively).

(2) supp(X1, T1) = X1 and supp(X2, T2) = X2.

(3) supp(Y1, S1) = Y1 and supp(Y2, S2) = Y2.

Proof. The equivalence of (1) and (3) is just [45, Theorem 7.6]. Thus it suffices to prove (3) ⇒ (2), as

(2) ⇒ (3) is direct. In fact, the direction of (3) ⇒ (2) follows directly from Theorem 7.10 (3). The proof

is completed.

7.3 Comments on the line by Glasner and Weiss

In order to consider the analogous notions in the relative setting, Glasner and Weiss introduced (see [31]):

Definition 7.15. Let π : (X,T ) → (Y, S) be a factor map between TDSs which is not a homeomor-

phism. π is called

(1) an entropy-generated extension if R(2)π is the ICER generated by (E2(X,T ) ∩R

(2)π ) ∪Δ2(X).

(2) a UPE extension if R(2)π \E2(X,T ) = Δ2(X).

(3) a CPE extension if htop(R,Z) > htop(T,X) (and so (Z,R) has positive relative topological entropy

with respect to (Y, S)) for any proper factor (Z,R) of (X,T ) with respect to (Y, S).

Similar to Theorem 7.10, we have (see [45]):


Proposition 7.16. Let π : (X,T ) → (Y, S) be a factor map between TDSs which is not a homeomor-

phism. Assume that π is an entropy-generated extension. Then

(1) Let μ ∈ M(X,T ). If

Eμ2 (X,T ) ∩R(2)

π = E2(X,T ) ∩R(2)π ,

then supp(μ) = π−1(supp(πμ)) and π−1(y) is a singleton for each y ∈ Y \supp(πμ).(2) π−1(y) is a singleton for each y ∈ Y \supp(Y, S).(3) supp(X,T ) = π−1(supp(Y, S)) and if π is weakly mixing, then supp(X,T ) = X.

Let π : (X,T ) → (Y, S) be a factor map between TDSs. By Proposition 6.10 and Proposition 7.9,

E2(X,T | π) ⊆ E2(X,T ) ∩R(2)π

and so if π has relative CPE, then π is an entropy-generated extension. Thus, Proposition 7.16 strengthens

Theorem 7.10. Moreover, by a similar argument, from the definitions it is not hard to obtain that each

CPE extension has relative CPE, each relative UPE extension is a UPE extension and all of them are

stronger properties than entropy-generated extension; while, unlike the absolute case, even with the

assumption of minimality on all considered TDSs, a UPE extension need not be a CPE extension and

vice versa (see [31]).

A TDS is mildly mixing if its product with any transitive TDS is transitive.

Recall that the map f : X → Y between topological spaces is semi-open if f(U) has a non-empty

interior whenever U is a non-empty open subset of X . Observe that each factor map between minimal

TDSs is semi-open (see [4]).

The following properties related to UPE extension were proved in [45].

Proposition 7.17. Let πi : (Xi, Ti) → (Yi, Si) be a factor map between TDSs which is a UPE extension

for each i = 1, . . . , n, n ∈ N.

(1) If all π1, . . . , πn are semi-open, then (X1 × · · · ×Xn, T1 × · · · × Tn) is transitive (weakly mixing,

mildly mixing, respectively) if and only if (Y1×· · ·×Yn, S1×· · ·×Sn) is transitive (weakly mixing, mildly

mixing, respectively).

(2) If supp(Yi, Si) = Yi for each i = 1, . . . , n, then π1 × · · · × πn is a UPE extension.

We should remark that the assumption of semi-openness is necessary in the above result which was

interpreted by [45, Remark 6.3]: there exists a factor map π : (X,T ) → (Y, S) such that π has relative

UPE (and so it is a UPE extension) and (Y, S) is a non-trivial transitive TDS while (X,T ) is not a

transitive TDS.

Acknowledgements I would like to thank Professors X. Ye and W. Huang for fruitful discussions and valuable

comments during the research. I also thank the referee for his careful reading the manuscript. This work

was supported by Foundation for the Authors of National Excellent Doctoral Dissertation of China (Grant No.

201018), National Natural Science Foundation of China (Grant No. 10801035) and Ministry of Education of

China (Grant No. 200802461004).

References

1 Adler R L, Konheim A G, McAndrew M H. Topological entropy. Trans Amer Math Soc, 1965, 114: 309–319

2 Ahn Y, Lee J, Park K K. Relative sequence entropy pairs for a measure and relative topological Kronecker factor. J

Korean Math Soc, 2005, 42: 857–869

3 Akin E, Glasner E, Huang W, et al. Sufficient conditions under which a transtive system is chaotic. Ergod Th &

Dynam Sys, 2010, 30: 1277–1310

4 Auslander J. Minimal Flows and Their Extensions. In: North-Holland Mathematics Studies, vol. 153. Amsterdam:

Elsevier Science Publishers B.V., 1988

5 Banks J, Brooks J, Cairns G, et al. On Devaney’s definition of chaos. Amer Math Monthly, 1992, 99: 332–334

6 Blanchard F. Fully positive topological entropy and topological mixing. Symbolic Dynamics and its Applications, AMS

Contemporary Mathematics, vol. 135. Providence, RI: Amer Math Soc, 1992, 95–105


7 Blanchard F. A disjointness theorem involving topological entropy. Bull Soc Math France, 1993, 121: 465–478

8 Blanchard F, Glasner E, Host B. A variation on the variational principle and applications to entropy pairs. Ergod Th

& Dynam Sys, 1997, 17: 29–43

9 Blanchard F, Glasner E, Kolyada S, et al. On Li-Yorke pairs. J Reine Angew Math, 2002, 547: 51–68

10 Blanchard F, Host B, Maass A, et al. Entropy pairs for a measure. Ergod Th & Dynam Sys, 1995, 15: 621–632

11 Blanchard F, Host B, Ruette S. Asymptotic pairs in positive-entropy systems. Ergod Th & Dynam Sys, 2002, 22:

671–686

12 Blanchard F, Huang W. Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin Dyn Syst, 2008, 20:

275–311

13 Blanchard F, Lacroix Y. Zero-entropy factors of topological flows. Proc Amer Math Soc, 1993, 119: 985–992

14 Bowen R. Entropy for group endomorphisms and homogeneous spaces. Trans Amer Math Soc, 1971, 153: 401–414

15 Boyle M, Fiebig D, Fiebig U. Residual entropy, conditional entropy and subshift covers. Forum Math, 2002, 14:

713–757

16 Bryant B F. On expansive homeomorphisms. Pacific J Math, 1960, 10: 1163–1167

17 Chung N P, Li H. Homoclinic group, IE group, and expansive algebraic actions. Preprint, 2011

18 Devaney R L. An Introduction to Chaotic Dynamical Systems (Second Edition). New York: Addison-Wesley, 1989

19 Dinaburg E I. A correction between topological and metric entropy. Dokl Akad Nauk SSSR, 1970, 190: 13–16

20 Dooley A, Zhang G H. Co-induction in dynamical systems. Ergod Th & Dynam Sys, in press

21 Dooley A, Zhang G H. Local entropy theory of a random dynamical system. Preprint, 2011

22 Dou D, Ye X, Zhang G H. Entropy sequences and maximal entropy sets. Nonlinearity, 2006, 19: 53–74

23 Downarowicz T, J Serafin. Fiber entropy and conditional variational principles in compact non-metrizable spaces.

Fund Math, 2002, 172: 217–247

24 Furstenberg H. The structure of distal flows. Amer J Math, 1963, 85: 477–515

25 Furstenberg H. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math

Systems Th, 1967, 1: 1–49

26 Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton University

Press, 1981

27 Glasner E. A simple characterization of the set of μ-entropy pairs and applications. Israel J Math, 1997, 102: 13–27

28 Glasner E. Ergodic Theory via Joinings. In: Mathematical Surveys and Monographs, vol. 101. Providence, RI: Amer

Math Soc, 2003

29 Glasner E, Thouvenot J P, Weiss B. Entropy theory without a past. Ergod Th & Dynam Sys, 2000, 20: 1355–1370

30 Glanser E, Weiss B. Strictly ergodic uniform positive entropy models. Bull Soc Math France, 1994, 122: 399–412

31 Glasner E, Weiss B. Topological entropy of extensions. Ergodic Theory and its Connections with Harmonic Analysis.

Cambridge: Cambridge University Press, 1995

32 Glasner E, Weiss B. On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems,

vol. 1B. Amsterdam: Elsevier B. V., 2006

33 Glasner E, Ye X. Local entropy theory. Ergod Th & Dynam Sys, 2009, 29: 321–356

34 Goodman T N T. Relating topological entropy and measure entropy. Bull London Math Soc, 1971, 3: 176–180

35 Goodwyn L W. Topological entropy bounds measure-theoretic entropy. Proc Amer Math Soc, 1969, 23: 679–688

36 Huang W. Stable sets and ε-stable sets in positive-entropy systems. Comm Math Phys, 2008, 279: 535–557

37 Huang W. Personal Communications, 2010

38 Huang W, Li S, Shao S, et al. Null systems and sequence entropy pairs. Ergod Th & Dynam Sys, 2003, 23: 1505–1523

39 Huang W, Maass A, Romagnoli P P, et al. Entropy pairs and a local Abramov formula for a measure theoretical

entropy of open covers. Ergod Th & Dynam Sys, 2004, 24: 1127–1153

40 Huang W, Maass A, Ye X. Sequence entropy pairs and complexity pairs for a measure. Ann Inst Fourier (Grenoble),

2004, 54: 1005–1028

41 Huang W, Ye X. Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology Appl, 2002, 117: 259–272

42 Huang W, Ye X. Topological complexity, return times and weak disjointness. Ergod Th & Dynam Sys, 2004, 24:

825–846

43 Huang W, Ye X. A local variational relation and applications. Israel J Math, 2006, 151: 237–279

44 Huang W, Ye X, Zhang G H. A local variational principle for conditional entropy. Ergod Th & Dynam Sys, 2006, 26:

219–245

45 Huang W, Ye X, Zhang G H. Relative entropy tuples, relative u.p.e. and c.p.e. extensions. Israel J Math, 2007, 158:

249–283

46 Huang W, Ye X, Zhang G H. Local entropy theory for a countable discrete amenable group action. J Funct Anal,

2011, 261: 1028–1082


47 Huang W, Ye X, Zhang G H. Lowering topological entropy over subsets. II. Preprint, 2011

48 Huang W, Ye X, Zhang G H. Lowering topological entropy over subsets. Ergod Th & Dynam Sys, 2010, 30: 181–209

49 Huang W, Yi Y. A local variational principle of pressure and its applications to equilibrium states. Israel J Math,

2007, 161: 249–283

50 Kerr D, Li H. Dynamical entropy in Banach spaces. Invent Math, 2005, 162: 649–686

51 Kerr D, Li H. Independence in topological and C∗-dynamics. Math Ann, 2007, 338: 869–926

52 Kerr D, Li H. Combinatorial independence in measurable dynamics. J Funct Anal, 2009, 256: 1341–1386

53 Kolmogorov A N. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces (in

Russian). Dokl Akad Sci SSSR, 1958, 119: 861–864

54 Kolyada S F. Li-Yorke sensitivity and other concepts of chaos. Ukrainian Math J, 2004, 56: 1242–1257

55 Kolyada S F, Snoha L. Some aspects of topological transitivity—a survey. Grazer Math Ber, 1997, 334: 3–35

56 Lemanczyk M, Siemaszko A. A note on the existence of a largest topological factor with zero entropy. Proc Amer

Math Soc, 2001, 129: 475–482

57 Ledrappier F, Walters P. A relativised variational principle for continuous transformations. J London Math Soc, 1977,

16: 568–576

58 Li T Y, Yorke J A. Period three implies chaos. Amer Math Monthly, 1975, 82: 985–992

59 Lind D, Schimidt K. Homoclinic points of algebraic Zd-actions. J Amer Math Soc, 1999, 12: 953–980

60 Ma X F, Chen E C, Zhang A H. A relative local variational principle for topological pressure. Sci China Math, 2010,

53: 1491–1506

61 Misiurewicz M. A short proof of the variational principle for a Zn+ action on a compact space. Asterisque, 1976, 40:

227–262

62 Mycielski J. Independent sets in topological algebras. Fund Math, 1964, 55: 139–147

63 Ornstein D S, Weiss B. Entropy and isomorphism theorems for actions of amenable groups. J d’Anal Math, 1987, 48:

1–141

64 Oprocha P, Zhang G H. On local aspects of topological weak mixing in dimension one and beyond. Studia Mathematica,

2011, 202: 261–288

65 Oprocha P, Zhang G H. On local aspects of topological weak mixing, sequence entropy and chaos. Preprint, 2010

66 Oprocha P, Zhang G H. On sets with recurrence properties, their topological structure and entropy. Topology Appl,

in press

67 Park K K, Siemaszko A. Relative topological Pinsker factors and entropy pairs. Monatsh Math, 2001, 134: 67–79

68 Rohlin V A. On the fundamental ideas of measure theory. Amer Math Soc Translation, 1952, 71: 1–54

69 Romagnoli P P. A local variational principle for the topological entropy. Ergod Th & Dynam Sys, 2003, 23: 1601–1610

70 Shapira U. Measure theoretical entropy for covers. Israel J Math, 2007, 158: 225–247

71 Song B, Ye X. A minimal completely positive entropy non-uniformly positive entropy example. J Difference Equ Appl,

2009, 15: 87–95

72 Tan F. The set of sequence entropies for graph maps. Topology Appl, 2011, 158: 533–541

73 Tan F, Ye X, Zhang R F. The set of sequence entropies for a given space. Nonlinearity, 2010, 23: 159–178

74 Walters P. An Introduction to Ergodic Theory. In: Graduate Texts in Mathematics, vol. 79. New York-Berlin:

Springer-Verlag, 1982

75 Weiss B. Strictly ergodic models for dynamical systems. Bull Amer Math Soc, 1985, 13: 143–146

76 Xiong J C, Yang Z G. Chaos caused by a topologically mixing map. Dynamical Systems and Related Topics (Nagoya,

1990). River Edge, NJ: World Sci Publ, 1991

77 Ye X, Zhang G H. Entropy points and applications. Trans Amer Math Soc, 2007, 359: 6167–6186

78 Zhang G H. Relative entropy, asymptotic pairs and chaos. J London Math Soc, 2006, 73: 157–172

79 Zhang G H. Relativization of complexity and sensitivity. Ergod Th & Dynam Sys, 2007, 27: 1349–1371

80 Zhang G H. Relativization and Localization of Dynamical Systems. Ph.D. Thesis, University of Science and Technology

of China, 2007 (available at http://homepage.fudan.edu.cn/zhanggh/files/2011/06/english.pdf)

81 Zhang G H. Variational principles of pressure. Discrete Contin Dyn Syst, 2009, 24: 1409–1435

82 Zimmer R J. Extensions of ergodic group actions. Illinois J Math, 1976, 20: 373–409

83 Zimmer R J. Ergodic actions with generalized discrete spectrum. Illinois J Math, 1976, 20: 555–588

Documents

Relativization of dynamical properties