Zero entropy continuous interval maps and MMLS-MMA
propertyNonlinearity
PAPER
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Yunping Jiang
Department of Mathematics, Queens College of the City University of
New York, Flushing, NY 11367-1597, United States of America
Department of Mathematics, Graduate School of the City University
of New York, 365 Fifth Avenue, New York, NY 10016, United States of
America
E-mail:
[email protected]
Received 25 September 2017, revised 7 February 2018 Accepted for
publication 9 March 2018 Published 2 May 2018
Recommended by Professor Masato Tsujii
Abstract We prove that the flow generated by any continuous
interval map with zero topological entropy is minimally
mean-attractable and minimally mean-L- stable. One of the
consequences is that any oscillating sequence is linearly disjoint
from all flows generated by all continuous interval maps with zero
topological entropy. In particular, the Möbius function is linearly
disjoint from all flows generated by all continuous interval maps
with zero topological entropy (Sarnak’s conjecture for continuous
interval maps). Another consequence is a non-trivial example of a
flow having discrete spectrum. We also define a log-uniform
oscillating sequence and show a result in ergodic theory for
comparison.
Keywords: minimally mean-attractable (MMA), minimally mean-L-stable
(MMLS), linear disjointness, Mobius function, oscillating sequence,
Sarnak’s conjecture, continuous interval map with zero topological
entropy Mathematics Subject Classification numbers: Primary; 11K65,
37A35, Secondary; 37A25, 11N05
Y Jiang
Printed in the UK
31
Nonlinearity
NON
10.1088/1361-6544/aab593
Paper
6
2689
2702
Nonlinearity
IOP
* This material is based upon work supported by the National
Science Foundation. It is also partially supported by a
collaboration grant from the Simons Foundation (grant number
523341) and PSC-CUNY awards and a grant from NSFC (grant number
11571122).
2018
1361-6544
Nonlinearity 31 (2018) 2689–2702
https://doi.org/10.1088/1361-6544/aab593
1. Introduction
Let C and R denote the complex plane and the real line. Let N = {1,
2, 3, · · · , n · · · } be the set of natural numbers. Suppose X is
a compact metric space with metric d(·, ·). Let C(X,C) be the space
of all continuous functions : X → C. Consider a continuous map f :
X → X . Then f generates a flow (i.e. dynamical system)
X = { f n : X → X}n∈{0}∪N
where f n means the nth iteration
f f · · · f n
and f 0 means the idenity. The Möbius function is the arithmetic
function µ(n) : N → {−1, 0, 1} defined as
µ(n) =
1 if n = 1; (−1)r if n = p1 · · · pr for r distinct prime numbers
pi; 0 if p2|n for a prime number p.
Sarnak proposed a conjecture which makes a connection between
number theory and ergodic theory. The conjecture can be stated
as
Conjecture 1 (Sarnak [14, 15]). The Möbius function is linearly
disjoint from (i.e. or- thogonal to) all flows with zero
topological entropy.
The linear disjointness can be defined more generally. Suppose c =
(cn)n∈N− is a sequence of complex numbers.
Definition 1 (Linear disjointness). We say c is linearly disjoint
from X if
lim N→∞
1 N
for any ∈ C(X,C) and any x ∈ X .
In order to understand conjecture 1 in ergodic theory, we gave the
following definition in [5].
Definition 2 (Oscillating Sequence). The sequence c is said to be
an oscillating sequence if
lim N→∞
1 N
with a technical condition
for some constant λ > 1.
An important example of an oscillating sequence is the Möbius
sequence u = (µ(n))n∈N due to Davenport’s theorem [2]. However,
there are many other oscillating sequences generated
Y Jiang Nonlinearity 31 (2018) 2689
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by arithmetic functions, for example, the Liouville function
defined as l(n) = (−1)(n), where (n) is the number of prime factors
of n, counted with multiplicity (see [1, 4]). There are
also many oscillating sequences from complex analysis and harmonic
analysis, for examples,
c = ( e2πin2α
( e2πicn log n
) for any fixed constant
c > 0 are oscillating sequences. One can find other examples of
oscillating sequences in [5]. Therefore, our new result theorem 2
and corollary 1 applies not only to Sarnak’s conjecture in number
theory corollary 2 but also to many other branches in mathematics
like ergodic theory, probability, and harmonic analysis.
The key point of conjecture 1 in the view of definition 1 is that
the convergence is for every point in the space. However, in
ergodic theory, the convergence is for almost all points for a
reference measure. To distinct our main result theorem 2 and
corollary 1 to those results in ergodic theory, we give a
definition of a log-uniform oscillating sequence and prove a result
in ergodic theory. This result in ergodic theory indicates why our
study of the MMA (minimally mean-attractable)-MMLS (minimally
mean-L-stable) property for flows is important.
Definition 3 (Log-uniform oscillating sequence). A sequence c is
said to be a log- uniform oscillating sequence if there are
constants A > 1 and B > 0 such that
max 0t<1
with also the technical condition (3).
An example of a log-uniform oscillating sequence is the Möbius
sequence u due to Davenport’s theorem [2].
Suppose B is the standard Borel σ-algebra on X (here we can relax
the compactness con- dition on X). Suppose ν is a Borel measure on
X. Then (X,B, ν) is a measurable space. A measurable and bijective
map f : (X,B, ν) → (X,B, ν) is called an automorphism if it is
measure-preserving, that is, ν( f−1(A)) = ν(A) for every A ∈ B (in
other words, we also called ν f-invariant). Let L1(X,B, ν) be the
space of all L1-functions on X.
Theorem 1. Suppose f : (X,B, ν) → (X,B, ν) is an automorphism.
Suppose c = (cn) is a log-uniform oscillating sequence. Then for
any φ ∈ L1(X,B, ν), we have that
lim N→∞
1 N
cn( f nx) = 0, for ν − a.e. x ∈ X. (5)
Since this theorem is not our main topic, just for a comparison, we
will prove it in the last section 3. In general, it is
difficult to get the convergence for every point in the above
theorem. That is a reason we need the following definitions.
Suppose E ⊂ N. The upper density D(E) of E in N is
D(E) = lim sup N→∞
N .
Definition 4 (MLS). A flow X is said to be mean-L-stable (MLS) if
for every ε > 0, there is a δ > 0 such that d(x, y) < δ
implies d( f nx, f ny) < ε for all n ∈ N except a subset E of
natural numbers with upper density D(E) less than ε.
Y Jiang Nonlinearity 31 (2018) 2689
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Mean-L-stable flows are an important class of dynamical systems in
ergodic theory. An example of a MLS flow is an equicontinuous
flow.
Definition 5 (Equicontinuity). A flow X is said to be
equicontinuous if for every ε > 0, there is a δ > 0 such that
d(x, y) < δ implies d( f nx, f ny) < ε for all n ∈ N.
We says a closed subset K ⊂ X is minimal (with respect to f) if f
(K) ⊂ K and if { f nx | n ∈ N} = K for every x ∈ K. We use K to
denote the sub-flow {( f |K)n : K → K}n∈{0}∪N. Again, for the
purpose of understanding conjecture 1 in ergodic theory, we gave
the following definitions in [5].
Definition 6 (MMLS). We say that a flow X is minimally MLS (MMLS)
if for every mini- mal subset K ⊆ X , the sub-flow K is MLS.
Definition 7 (MMA). Suppose X is a flow. Suppose K ⊂ X is minimal.
We say x ∈ X is mean-attracted to K if for any ε > 0 there is a
point z = zε,x ∈ K (depending on x and ε) such that
lim sup N→∞
d( f nx, f nz) < ε. (6)
The basin of mean-attraction of K, denoted Basin(K), is defined to
be the set of all points x which are mean-attracted to K. It is
trivial that K ⊂ Basin(K). We call X minimally mean- attractable
(MMA) if
X = K
Basin(K) (7)
where K varies among all minimal subsets of X.
Recall that a point x ∈ X is attracted to K if
lim n→∞
d( f nx, K) = 0.
In general, this does not imply to that x is mean-attracted to K.
We proved the following theorem in [5]. The reader is encouraged to
compare this result
with theorem 1.
Theorem 1 (MMA/MMLS and disjointness). Any oscillating sequence c
is linearly disjoint from all MMA and MMLS flows X . Moreover, the
limit in (1) is uniform on every minimal subset K.
Since a MMA and MMLS flow has zero topological entropy (for
example, this follows from [9]), this theorem confirms Sarnak’s
conjecture for all MMA and MMLS flows. We provided some examples of
flows which are MMA and MMLS in [5]. However, these exam- ples
except for Denjoy counter-examples are equicontinuous when they are
restricted on their minimal subsets. It is an interesting problem
to find further examples which are MMA and MMLS but not
equicontinuous when they are restricted on their minimal subsets.
We give a complete answer to this problem for continuous interval
maps with zero topological entropy in this paper. The main result
in this paper is that
Theorem 2 (Main theorem). Suppose I = [a, b] ⊂ R is a closed
interval. Suppose f : I → I is a continuous map with zero
topological entropy. Then the flow X generated by f is a MMA and
MMLS flow.
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We will prove this theorem in section 2. Now let us discuss
some very important conse- quences from our main result theorem 2.
The first one is our main goal in this area.
Corollary 1 (Continuous interval map and disjointness). Suppose I =
[a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous
map with zero topological entropy. Then any oscillating sequence c
is linearly disjoint from the flow X generated by f.
The proof of corollary 1 is given by combining the proof of theorem
2 and the proof of theorem 1. This proof provides a completely new
proof of Sarnak’s conjecture for all flows generated by continuous
interval maps with zero topological entropy (see [8] for a
different proof and remark 1 for another approach) as
follows.
Corollary 2 (Continuous interval map and Möbius function). Suppose
I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a
continuous map with zero topologi- cal entropy. The Möbius function
is linearly disjoint from the flow X generated by f.
Let us also discuss some consequence of our main result theorem 2
in ergodic theory. Suppose K ⊂ X is a subspace. Let BK be the
standard Borel σ-algebra on K and let ν be a Borel probability
measure on K. Let L2(K,BK , ν) be the space of all L2 functions on
K.
Suppose f is a measure-preserving map from (K,BK , ν) into itself.
A complex number λ is called an eigenvalue if there is a function φ
∈ L2(K,BK , ν) such that φ( fx) = λφ(x) for ν-a.e. x ∈ K. Here φ is
called an eigenfunction with respect to λ. We say f : (K,BK , ν) →
(K,BK , ν) has discrete spectrum or pure point spectrum if there
exists an orthonormal L2(K,BK , ν)-basis consisting of
eigenfunctions. Theorem 2 provides a non-trivial example of flows
having dis- crete spectrum (see [9, theorem 3.8]).
Corollary 3 (Continuous interval map and discrete spectrum).
Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a
continuous map with zero topo- logical entropy. Suppose X is the
flow generated by f. Suppose K ⊆ X is a minimal subset with respect
to f. Then for every ergodic f-invariant measure ν on K, f : (K,B,
ν) → (K,B, ν) is a measure-preserving map having discrete
spectrum.
Remark 1. Corollary 3 combining with the work in [11] (see also
[6]) gives another proof of Sarnak’s conjecture for all flows
generated by continuous interval maps with zero topologi- cal
entropy as present in corollary 2.
Remark 2. I would like to point out that there are flows generated
by higher-dimensional maps with zero topological entropy which are
not MLS when they are restricted on their mini- mal subsets. Thus
in the higher-dimensional case, the oscillation property alone is
not enough. We need instead the higher-order oscillation property.
The reader who is interested in the higher-order oscillation
property for a sequence c and the role it plays in Sarnak’s
conjecture for higher-dimensional dynamical systems is encouraged
to read my paper [7].
2. Proof of the main result theorem 2
Suppose I = [a, b] ⊂ R is a closed interval and f : I → I is a
continuous map. Suppose the topological entropy ent( f ) = 0. In
other words, this means that f cannot have any horseshoe- type
invariant subset in I. More precisely, this means that f can not
have a forward f-invariant closed subset X ⊂ I such that f : X → X
is topologically conjugate (or semi-conjugate) to a sub-shift of
finite type σA : ΣA → ΣA with the maximal Lyapunov exponent greater
than 0.
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For the sub-shift of finite type σA : ΣA → ΣA, we mean that A =
(aij)d×d is a d × d-square matrix of 0’s and 1’s,
ΣA = {w = i0i1 · · · in−1in · · · | in−1 ∈ {1, 2, · · · , d},
ain−1in = 1, n ∈ N}
and
σA(w) = i1 · · · in−1in · · · .
By abuse of notation, we use ΣA to mean a horseshoe-type forward
f-invariant subset X ⊂ I . Since A is a nonnegative square matrix,
from the Perron–Frobenius theorem, it has a maximal nonnegative
eigenvalue eA. If A is also irreducible, eA is simple, unique, and
positive. The statement that maximal Lyapunov exponent of σA is
greater than 0 is equivalent to saying that eA > 1.
A point p ∈ X is called a periodic point of period n 1 if fn( p ) =
p. The minimal such n is called the primitive period. Following the
proof of Sharkovskii’s theorem (for example, the reader may refer
to book [3] or book [13, chapter 4]), we see that if f has a
periodic point whose period is different from all powers of 2, then
f has a horseshoe-type invariant subset. For example, if f has a
periodic point of period 3, then f has a horseshoe-type invariant
subset σA : ΣA → ΣA for
A =
5)/2. Thus we have that
Lemma 1. The period of any periodic point of f is 2n for some n
0.
For any x ∈ I , let
ω(x) = ∩∞ n=0{ f kx | k n},
be the ω-limit set of x under f. Let K = ω(x). Consider the
sub-flow
K = {( f |K)n : K → K}n∈{0}∪N. (8)
It is a minimal flow (refer to [13] or the argument after lemma 2).
We have two cases:
(I) the set K is finite and (II) the set K is infinite.
Case (I) is easier. In this case,
K = { p0, p1, · · · , p2n−1}
is a periodic orbit of period 2n for some n 0. That is, f ( pi) =
pi+1 (mod 2n) for i = 0, 1, · · · , 2n − 1. Therefore, the
following lemma is easily verified.
Lemma 2. In Case (I), the sub-flow (8) is equicontinuous, thus,
MLS. Moreover, x is mean-attractable to K.
Case (II) is more complicated. Since the topological entropy of f
is zero, in this case, K con- tains no periodic point (see, for
example, [13, chapter 5]). Since f cannot have any horseshoe- type
forward f-invariant subset in I, we can find a sequence of
sets
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of closed intervals In,k such that
(1) for each n 1, intervals in ηn are pairwise disjoint; (2) f
(In,k) = In,k+1 (mod 2n) for any n 1 and 0 k 2n − 1; (3) for each n
1 and In,k ∈ ηn, two and only two intervals In+1,k and In+1,k+2n in
ηn+1 are
sub-intervals of In,k, that is,
In+1,k ∪ In+1,k+2n ⊆ In,k;
(4) K ⊆ ∩∞ n=1 ∪
2n−1 k=0 In,k .
This is a well-known fact about continuous interval maps with zero
topological entropy among all experts in this field (for example,
refer to [13, Chapter 5]).
We put a symbolic coding on each interval In,k in ηn for n 1 and 0
k 2n − 1. Consider the binary expansion of k as
k = i0 + i12 + · · · im2m + · · ·+ in−12n−1
where i0, i1, · · · , im, · · · , in−1 = 0 or 1. Let us only
remember the coding wn = i0i1 · · · im · · · in−1 and label In,k by
wn, that is, In,k = Iwn.
Define the space of codings
Σn = {wn = i0i1 · · · im · · · in−1 | im ∈ {0, 1}, m = 0, 1, · · ·
, n − 1}
with the product topology. We have a natural shift map
σn(wn) = i1 · · · im · · · in−1 : Σn → Σn−1.
It is a continuous 2-to-1 map. Then we have an inverse limit
system
{(σn : Σn → Σn−1); n ∈ N}.
Let
(σ : Σ → Σ) = lim ←−
(σn : Σn → Σn−1).
Explicitly, we have that
Σ = {w = i0i1 · · · in−1 · · · | in−1 ∈ {0, 1}, n ∈ N}
and
Consider the metric d(·, ·) on Σ defined as
d(w, w′) =
∞∑ n=1
|in−1 − i′n−1| 2n
for w = i0i1 · · · in−1 · · · and w′ = i′0i′1 · · · i′n−1 · · ·. It
induces the same topology as the topology from the inverse limit
and makes Σ a compact metric space.
On Σ, we have a map called the adding machine denoted as add and
defined as follows: for any w = i0i1 · · · ∈ Σ, consider the formal
binary expansion a =
∑∞ n=1 in−12n−1, then
a + 1 = ∑∞
n=1 i′n−12n−1 has a unique formal binary expansion. Here ‘a + 1’ is
just a notation. It means if i0 = 0, then i0 = 1 and all other i′n
= in and if i0 = 1, then i0 = 0 and then consider i1 + 1, and so
on. The adding machine is
Y Jiang Nonlinearity 31 (2018) 2689
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add(w) = i′0i′1 · · · i′n−1 · · · : Σ → Σ.
It is a homeomorphism of Σ to itself. Moreover, we have
Lemma 3. The flow
is minimal and equicontinuous.
We leave the proof for the reader since it is not difficult. For a
point w = i0i1 · · · in−1 · · · ∈ Σ, let wn = i0i1 · · · in−1. We
use
[w]n = {w′ ∈ Σ | w′ n = wn}
to denote the n-cylinder containing w. One can check that for every
w ∈ Σ,
· · · ⊂ Iwn ⊂ Iwn−1 ⊂ · · · ⊂ Iw2 ⊂ Iw1 ,
is a nested sequence of closed intervals. Therefore
Iw = ∩∞ n=1Iwn
is a non-empty, connected, and compact subset of I. Moreover, we
have that f (Iw) = Iadd(w) and
K ⊂ ∪w∈ΣIw.
The set C = {[w]n | w ∈ Σ, n ∈ N} of all n-cylinders forms a
topological basis for Σ. Let B be the σ-algebra generated by C. We
have a standard probability measure ν defined on B such that
ν([w]n) = 1/2n for all w ∈ Σ. We say ν is add-invariant if
ν(add−1(A)) = ν(A) for all A ∈ B. We say ν is ergodic if add−1(A) =
A for A ∈ B implies either ν(A) = 0 or 1. The following lemma is
also well-known among experts in this field and not difficult to be
verified. We also leave it to the reader.
Lemma 4. The measure ν is add-invariant and ergodic.
In Case (II), we consider two different cases:
(a) there is a point w0 ∈ Σ such that Iw0 = {xw0} contains only one
point or (b) all Iw for w ∈ Σ are closed intervals.
In Case (II)(a), we have that ω(x) = ω(xw0). Without loss of
generality, we assume x = xw0 and every Iw = {xw} contains only one
point. Let
τn = max wn∈Σn
|Iwn |.
We claim that τn → 0 as n → ∞. We prove it by contradiction. If
there is a subsequence {nk}∞k=1 of the natural numbers and a number
ε0 > 0 such that τnk ε0, then we have an inter- val Iwnk
∈ ηnk such that |Iwnk | ε0. Since Σ is a compact metric space, we
have a convergent
subsequence of {wnk}∞k=1 in Σ. Suppose {wnk}∞k=1 is itself a
convergent sequence with the limit point w ∈ Σ, then the interval
Iwnk
converges to an interval Iw as k → ∞. The length |Iw| is greater
than ε0. This contradicts the fact Iw contains only one point. This
proves the claim.
Lemma 5. In Case (II)(a), the sub-flow K in (8) is equicontinuous,
thus, MLS.
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Proof. We can set a one-to-one and onto correspondence h(w) = xw
from Σ to K. Since τn → 0 as n → ∞, for any ε > 0, there is a N
> 0 such that τN < ε. For the given N, we can find a δ > 0
such that for any w, w′ ∈ Σ with d(w, w′) < δ , then wN =
w′
N . This implies that both points xw and xw are in the same
interval IwN. So |h(w)− h(w′)| τN < ε. This proves that h is
continuous. Since both Σ and K are compact, we have that h−1 is
also continuous. Thus h is a homeomorphism. Since f |K = h add h−1
and since the flow A is equicontinu- ous, the sub-flow K is
equicontinuous.
Lemma 6. In Case (II)(a), x is mean-attracted to K.
Proof. Since K ⊂ ∩∞ n=1 ∪wn∈Σn Iwn and since τn → 0 as n → ∞, for
any ε > 0, there is
an integer n0 > 0 such that τn0 < ε. Then we have an integer
m0 > 0 such that f m0 x ∈ Iwn0
for some w ∈ Σ. Since K ∩ Iwn0 = ∅ and f |K = h add h−1 is a
homeomorphism, we have
z ∈ K such that f m0 z ∈ K ∩ Iwn0 . This implies that
| f m0 x − f m0 z| τn0 < ε.
Since f m0+n(Iwn0 ) ∈ ηn0 for all n 0, we have that
| f m0+nx − f m0+nz| τn0 < ε
for all n 0. This implies that
lim sup N→∞
We proved that x is mean-attracted to K.
In Case (II)(b), we consider a fixed interval Iw0. Let
ε0 = |Iw0 | > 0.
The orbit {addn(w0) | n ∈ Z} is dense in Σ. Then we can find two
points u, v ∈ K such that the distance |u − v| is arbitrary small
and such that there are infinitely many positive integers n such
that Iadd−n(w0) ⊂ Iuv. Then we have
f n(Iuv) ⊃ f n(Iadd−n(w0)) = Iw0 .
Thus | f nu − f nv| ε0 for infinitely many positive integers n.
This proves that
Lemma 7. In Case (II)(b), the sub-flow K in (8) is not
equicontinuous.
The main work in this paper is to prove the sub-flow K in (8) is
still MLS in Case (II)(b).
Lemma 8. In Case (II)(b), the sub-flow (8) is MLS.
Proof. Since the probability measure ν is add-invariant and
ergodic, the Birkhoff ergodic theorem implies that
lim N→∞
= lim N→∞
= ν(A),
2698
for any A ∈ B and ν-a.e. w ∈ Σ. For any ε > 0, since Iw ∩ Iw′ =
∅ for all w = w′ ∈ Σ and since
∑ w∈Σ |Iw| |I| < ∞,
there are an integer N0 > |I|/ε+ 1 and 0 m0 |I|/ε+ 1 points,
denoted as wj, 1 j m0, in Σ such that for any w = w j ∈ Σ for all 1
j m0 and n N0, |Iwn | < ε.
Let N1 > N0 be an integer such that ν([w]N1) < ε/m0 for all w
∈ Σ. We have a number δ > 0 such that for any u, v ∈ K with |u −
v| < δ, the interval [u, v] ⊂ Iwu,v
N1 for some wu,v ∈ Σ.
For any given |u − v| < δ and for each 1 j m0, define
Ej = {k ∈ N | add−k([w j]N1) = [wu,v]N1}.
From the Birkhoff ergodic theorem (see the formula in the beginning
of the proof), we have that
lim N→∞
lim N→∞
m0∑ j=1
#(Ej ∩ [1, N])
N < ε. (10)
Thus we have that the upper density D(E) is less than ε. For any n
∈ N \ E , addn([wu,v]N1) = [w j]N1 for all 1 j m0. This implies f
n(u),
f n(v) ∈ IwN1 for w = w j for all 1 j m0. Thus,
| f n(u)− f n(v)| < ε, ∀ n ∈ N \ E. (11)
Both (10) and (11) imply that f : K → K is MLS. We completed the
proof.
Lemma 9. In Case (II)(b), x is mean-attracted to K.
Proof. We use the same notation as that in the proof of lemma 8.
For any ε > 0, since Iw ∩ Iw′ = ∅ for all w = w′ ∈ Σ and
since
∑ w∈Σ |Iw| |I| < ∞, there are an integer
N0 > |I|/ε+ 1 and 0 m0 |I|/ε+ 1 points, denoted as wj, 1 j m0,
in Σ such that for any w = w j ∈ Σ for all 1 j m0 and n N0, |Iwn |
< ε/3.
Let N1 > N0 be an integer such that ν([w]N1) < ε/(3|I|m0) for
all w ∈ Σ. Fix an inter- val Iw′
N1 with length less than ε. There is an integer k 0 such that f kx
∈ Iw′
N1 . Take
z0 ∈ Iw′ N1 ∩ K . Then we have a point z = zx,ε ∈
I(add−k(w′))N1
∩ K such that f kz = z0. Thus we have that | f kx − f kz| <
ε.
Define
Ej = {k ∈ N | add−k([w j]N1) = [w′]N1}.
From the Birkhoff ergodic theorem (see the formula in the beginning
of the proof of lemma 8), we have that
lim N→∞
2699
lim N→∞
k∑ n=1
| f k+nx − f k+nz|+ 1 N
∑ n∈N,k+n∈E
= I + II + III.
First for N large, we have I < ε/3. Secondly, for any n ∈ N such
that k + n ∈ N \ E,
[w]N1 = addk+n([w′]N1) = [w j]N1 .
This implies f k+nx, f k+nz ∈ IwN1 for w = w j for all 1 j m0.
Thus,
| f k+nx − f k+nz| < ε
3 , ∀ n ∈ N and k + n ∈ N \ E.
This implies that II < ε/3.
Finally, since | f k+nx − f k+nz| |I| is always true, we have
III #(E ∩ [1, N])|I|
N .
The upper density of E given in (13) implies that
lim sup N→∞
This proves that x is mean-attracted to K.
Lemmas 2, 5, 6, 8 and 9 complete the proof of theorem 2.
Remark 3. The proof given in this section not only works for
continuous interval maps with zero topological entropy but also
works for any dynamical system f : X → X of a com- pact space X
satisfying that (i) it is MMA and (ii) when f is restricted to any
minimal subset K ⊆ X , f : K → K is semi-conjugate to the adding
machine add : Σ → Σ, that is, there is a continuous surjective map
h : K → Σ such that h f = add h. Jian Li sent me a note after the
first version of this paper was appeared in ArXiv and mentioned
that they are working
Y Jiang Nonlinearity 31 (2018) 2689
2700
on a similar result as stated in corollary 2 for graph maps with
zero topological entropy by using a different approach [10]. Graph
maps with zero topological entropy and circle ho- meomorphisms and
continuous interval maps with zero topological entropy are
interesting examples satisfying (i) and (ii). Thus our proof in
this section could be used to prove not only continuous
interval maps with zero topological entropy but also graph maps
with zero topo- logical entropy and circle homeomorphisms. It is
still not clear to me if there is an example of a dynamical system
f : X → X of a compact space X which satisfies only (i) but not
(ii), however, it is still MMLS.
3. Proof of theorem 1
Without loss of generality, we assume that ν is ergodic. Otherwise,
we consider on every ergodic component one by one. For any φ ∈
L1(X,B, ν), we define
σφ(n) = ∫
∫
These numbers character a measure σφ on the unit circle
T = {e2πit | 0 t < 1}
∫
SN(x) = 1 N
N∑ n=1
cnφ( f nx).
Since c = (cn) is a log-uniform oscillating sequence, we have two
constants A > 1 and B > 0 such that
SN2 = (∫
, ∀ N 2.
For any r > 1, let [rm] be the largest integer less than or
equal to rm. Then we have that
S[rm]2 B
This implies that
S[rm]2 B
S[rm](x) = 1
cnφ( f nx) → 0 as m → ∞ for ν − a.e.x ∈ X.
Y Jiang Nonlinearity 31 (2018) 2689
2701
For any integer N > 0, we have a unique integer m > 0 such
that [rm] N < [rm] + 1 [rm+1], then we have that
|SN(x)| = 1 N
N∑ n=1
1 N
N∑ [rm]+1
cnφ( f nx) .
Let us first prove the theorem under the assumption that φ ∈
L∞(X,B, ν). Under this assumption, let λ′ > 1 be the dual number
of λ in the technical condition (3), that is, 1/λ+ 1/λ′ = 1. The
Hölder inequality implies that
1 N
λ′ .
1 λ′ as m → ∞.
Therefore, for any ε > 0, we have r > 1 such that Cφ∞(r − 1)
< ε/4. For this fixed r, we have a m0 > 0 such that
C 1 λ φ∞
1 λ′
4 , ∀ m > m0,
and |S[rm](x)| < ε/2 for ν-a.e. x ∈ X for all m > m0. Thus we
get that
|SN(x)| < ε for ν − a.e. x ∈ X and m > m0.
We proved that
SN(x) → 0 as m → ∞ for ν − a.e. x ∈ X
when φ ∈ L∞(X,B, ν). In general, we find a sequence {φm}∞m=1 ⊂
L∞(X,B, ν) such that
φ− φm1 =
Note that
lim N→∞
Y Jiang Nonlinearity 31 (2018) 2689
2702
For any ε > 0, we have m > 0 such that φ− φm1 < ε/2. For
this fixed m, from the first part of the proof, we have a N0 > 0
such that
IN,m(x), IIN,m(x) < ε
Thus we get SN(x)
≤ ε, for ν-a.e. x ∈ X and any N > N0.
Therefore, SN(x) → 0 as N → ∞ for ν-a.e. x ∈ X . This completes the
proof.
Acknowledgment
This paper was completed during my visit to the National Center for
Theoretical Sciences (NCTS) at National Taiwan University in 2016.
I would like to thank NCTS for its hospitality. I would like to
thank my student John Adamski for his help on revising this
paper.
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Y Jiang Nonlinearity 31 (2018) 2689
Abstract
3. Proof of theorem 1
Acknowledgment
References