Periodic Point Free Homeomorphism of T2Author(s): Michael HandelSource: Proceedings of the American Mathematical Society, Vol. 107, No. 2 (Oct., 1989), pp.511-515Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2047842 .

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 107, Number 2, October 1989

PERIODIC POINT FREE HOMEOMORPHISM OF T2

MICHAEL HANDEL

(Communicated by Kenneth R. Meyer)

ABSTRACT. Suppose that f: T2 -- T2 is an orientation preserving homeomor- phism of the torus that is homotopic to the identity and that has no periodic points. We show that there is a direction 0 and a number p such that every orbit of f has rotation number p in the direction 0 .

1. INTRODUCTION

The rotation number of an orientation preserving circle homeomorphism is an essential tool for analyzing its dynamics. For orientation preserving homeo- morphisms f: Tn -, Tn (n > 1) of higher dimensional tori that are homotopic to the identity, there is a straightforward generalization of the rotation number that we may consider ([He]). Namely, for each x E Tn, choose lifts x E R

n n and f: Rn - Rn to the universal cover and define the translation vector -r(f, x) to be the limit, if it exists, of the average displacement vector (fn (x) - x)/n. If r(f, x) is well-defined, then the rotation vector p(f, x) E Tn of the f-orbit of x is the projected image of z(f,x).

The rotation number can also be computed as an average p(f, x) =

limn l Ein0 ~ (f'(x)) where ~o: Tn Tn is the displacement function (0(y) =

f(y) -y. The ergodic theorem therefore implies that p(f, x) is defined on a set of full ,u-measure for any f-invariant measure ,u. Moreover, if f is uniquely ergodic then p(f, x) is defined for all x and in fact is independent of x .

Herman [He] has shown that there are orientation preserving minimal diffeo- morphisms f: Tn -, Tn (n > 3) for which not every p(f, x) is well defined. Thus any hope of finding a sufficient topological condition to guarantee the ex- istence of a rotation vector at each point in Tn is restricted to the case n = 2.

In this paper we show that if f: T2 - T2 is orientation preserving, homo- topic to the identity and periodic point free (see Remark 1.3 for a weakening of this condition) then "half" of a rotation vector is defined. More precisely, for any vector vi E R , define -r(f, , vi) to be the limit, if it exists, of the pro- jection of the average displacement vector (ff(x) - k)/n onto vi. If -r(f, f) is well-defined and vi is a unit vector then r(f, x , vi) equals the scalar product

Received by the editors July 3, 1988 and, in revised form, October 5, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 54H20, 57M99.

? 1989 American Mathematical Society 0002-9939/89 $1.00 + S.25 per page

511

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512 MICHAEL HANDEL

Tr(f, X) *V. Converselyif r(f, Y , x'J) and Tr(f k , V2 ) are defined for linearly independent 131 and v2' then r(f , x) is well-defined and uniquely determined by T(f,X V ) and T(f, X, V2)

2 2.

Theorem 1.1. If f: T 2-+ T2 is an orientation preserving periodic point free homeomorphism that is homotopic to the identity, then there exists a vector v E R such that T(f ,x,v) is defined for all lifts f:R -+ R and E R; moreover T(f, C , xv) is independent of x .

Remark 1.2. Skew products f(0 , 02) = (0I + a , g(01 , 02)) where a E S1 and g: T2 -+ S are good examples to keep in mind. In this case vi is tangent to the

2 line that projects onto the 0 1-circle in T' and f(f k , xv) mod 1 = a . Herman [He] has shown that p(f , x) always exists for these f . If a is irrational then p(f , x) is independent of x but if a is rational then p(f , x) may vary.

Remark 1.3. Theorem 1.1 is a special case of a more general result about zero entropy surface diffeomorphisms. Using techniques similar to those in [Ha, Section 9], one can show for example that the conclusions of Theorem 1.1 remain valid if one only assumes that f has zero topological entropy and a finite but non-empty fixed point set. We have concentrated on the periodic point free case in this paper because the techniques are very different from those of the zero entropy case and because, as the following example (shown to the author by A. B. Katok) demonstrates, rotation vectors do not always exist if f has fixed (or periodic) points. It is an open question as to whether or not p(f, x) is defined for all x when f is periodic point free.

Example 1.4. Let Y be a unit vector field on T2 that generates an irrational flow in the direction w'. Choose a point P E T2 and a function u: T2 - [0, 1] such that u- (0) = P and such that the flow pt generated by u Y is not uniquely ergodic. In particular, there is an f-invariant ergodic measure ,u such that fud4u>O.

Let S = {x E T2:D(x) = lim,o T (())dt is well defined}. The ergodic theorem implies that A = {x E S: ?(x) > (f u du)/2} is non-empty. The set B = {x E S: D(x) = 0} is also non-empty since it contains the half- infinite flow line whose forward end is P. As 'D is constant on flow lines, A

2 2 2- an and B are both dense in T . It follows that S 77 T . For y E T S and lifts y of y and f of f = 1, T(f, y, w) is not defined.

2. PROOF OF THEOREM 1. 1

Let T1 (x,y) = (x + 1 ,y) and T2(x, y) = (x,y + 1) be the covering transla- tions of R 2; Let P1 ,P2: R 2-+ R2 be the projections onto the first and second coordinate factors respectively.

2 2 For any lift g: R -- R of an orientation preserving homeomorphism g: T727 T2, wedefinefunctions a (k, ,{nk}) and fl (4,,{nk}) where xeR2

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HOMEOMORPHISM OF T2 513

and where {nk } is a sequence of positive integers such that both limits

a(, x, '{nk}) = lim (p l k (.x)-p (x))/nk

and (k, xc, {nkl) = lm (p2fkn (.x) -p2(iT))/nk

exist. If either a(k,J,{nk}) : 0 or fi(k, ,{nk}) : 0, then we may define

0(4 , , {nk}) to be the polar coordinate angle of the vector (a(4 , , {nk}),

/3(4'k, {nk})). Define 9(g) to be the union of all 0(4,x,{nk}) as x and {nk} vary. If x projects to a periodic point of g, then 0(4 x, {nk}) is inde- pendent of {nk } and we will sometimes write 0(4, x) in this circumstance.

The following proposition was inspired by an argument of Franks [Fr].

Proposition 2.1. If 4: R2 - R2 is fixed point free, then 9(4) is contained in a semi-circle.

Proof. Assume that 4 is fixed point free. Fix 0 = 0(4,k,{nk}) E 0(4) and O < c < minkER2 I4g() - x I. After passing to a subsequence of {nk} we may

assume that Ignk (x) _ gnl (x) I < e for all k and / where x E T2 is the projected

image of x. Choose k > / and let g,: T2 - T2 be a homeomorphism that is e-close to the identity, that carries gnk (X) to gnf (x) and that fixes g1(x) for n < i < nk. Then h = g, o g is e-close to g in the C -topology and there is

a lift h that is c-close to 4 in the C0-topology. In particular h is fixed point free. The point y = gnf (x) is h-periodic and if k is sufficiently large relative to 1, then 0(h , y) is e-close to 0.

This procedure can be applied simultaneously to any three points 61, 02 and 3 in 0(4). (If 61 = 0(4,i l,{nk}) and 02 = 0( ,i2,{ml}) where xl and

2 x2 project onto the same point x E T2, then y1 and Y2 are obtained by clos- ing distinct g-orbits of points that are close to x.) Thus for all c > 0 and all

o1' 02, 03 E e(4), there exists a homeomorphism h: T2 - T2 that is c-close to g in the C0-topology and that has periodic orbits Y1, Y2 and y3 satisfy-

ing 60(h , ji) - Oil < e. Radially conjugate h: R 2- R2 to a homeomorphism

H: intD2 - intD2 , and label by z1 the points in intD2 corresponding to Yi under the conjugating map. Extend H over aD2 by the identity. Then the hypotheses of Lemma 2.2 below are satisfied with coi and ai being antipodal points and with I1coi - OiI < e . We conclude that the Oi 's and (6i + 7r) 's do not

alternate on S1 and hence that the 60's lie in a semi-circle. o

Lemma 2.2. Suppose that H: D2 - D2 is an orientation preserving homeomor- phism of the closed disk and that Y1, Y2, y3 E int D2 have a-limit sets a1 , a2 , a3

2 and co-limits sets co1, Ic2, C03 that are single points on aD . If the ai 's and

co 's alternate around aD2 then Hlint D2 has a fixed point.

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514 MICHAEL HANDEL

The proof of Lemma 2.2 is contained in [Ha, Section 5 (Proposition 5.4)]. Recall that the Brouwer translation theorem implies that if h: R2 - R2 is a fixed point free orientation preserving homeomorphism then each h-orbit is without accumulation points. One may intuitively think of Lemma 2.2 as stating that if h: R2 - R2 has orbits that move in directions that do not all lie in some half-plane, then some orbit of h has bounded forward orbit.

2 2 2 2 Proof of Theorem 1.1 . Choose a lift f: R -* R of f: T -* T . We will show that there are constants A , B and C such that

(A, B) *(a>(f, x, nk}) , fl(f x, {cnk})

=Aa(f,x,{nk}) +Bfl(f,c{nk}) = C

for all x and { nk } (for which a (f , x, { nkk}) and f,(f,x{n k }) are defined). This implies that r(f , xv) is well-defined and independent of x where vi is the vector (A, B) .

The functions a and ,B behave well when f is replaced by any lift of any positive iterate of f . Namely,

a(T7Tjfl2 , , {[n /q]}) = qa(f, , {nk })-p

and /ST 2 , [n k1q]j) = q,B(f, x., nk })-

where [] is the greatest integer function and q > 0. If fl(f, x, {nk}) is independent of x and {nk}, choose A = 0, B = 1 and

C = fl(f,n, {nk }). Otherwise we may assume that ,B takes on at least two values fl, = fl(f,i1,f{ui}) < fl(f, 2,{wj}) = fl2. Replacing f by 2 where fl, < r/q < f32 if necessary, we may assume that f1? < 0 < fl2. Define a1 = a(f, j,{u1}) and a2 = a(j 2,{wi})-

As p/q -Y = (fi2a1 - fl1a2)/(f2 - fl1),6(T7 Pfq1,1,{[uj/q]}) and 6(T f P1'k2 {[wj/q]}) approach distinct (because 2 > 0 > Al) angles ( and (2 whose tangents fll/(a,-y) and f2/(a2-y) are equal. Thus q0 and (2

are antipodal. The points O(T Pfq," x , {[uj/q]}) and O(T f" , X2, {[Wj /q]}) lie in the interior of the semi-circle that has endpoints {VI 1 ( 2} and that con- tains 0 for p/q > y and in the interior of the semi-circle that has endpoints

(P I 2} and that contains 7r for p/q < y. Choose any x and {nk} such that a(f ,,{nk}) and fl(f,Yc,{nk}) are

defined. If a(f,x,{nk}) $ y or if fl(f''{nk}) :$ 0, then 0(T Pfj,x, {[ni/q]}) is defined for all p/q close to y and is a uniformly continuous func- tion of p/q. Applying Proposition 2.1 to p/q close to and on both sides of y, we conclude that o = limp/q y 0(TPf , t , {[n /qc}) is either (p1 or (. In particular,

tango = (,B(f, ,{nk})/(a(f x, *{nk}) - y)

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HOMEOMORPHISM OF T2 515

is independent of x and {nk}. Choose A = tan (o,B = -1 and C = ytan( . Note that Aa(f , ,{nk})+ Bf(f ,i,{nk}) = C even when a (fJ, {nk}) =

and fi(fj, {nk})=0*

REFERENCES

[Fr] Franks, John, Recurrence and fixed points of surface homeomorphisms, to appear in Ergodic Theory and Dynamical Systems.

[Ha] Handel, M., Zero entropy surface diffeomorphisms, preprint . [He] Herman, M., Une methode pour minorer les exposants de Lyapounov et quelques exemples

montrant le caractere local d'un theoreme d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), no. 3, 453-502.

DEPARTMENT OF MATHEMATICS, LEHMAN COLLEGE, BRONX, NEW YORK 10468

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