A simple computable criteria for the existence of horseshoes

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A simple computable criteria for the existence of horseshoes

Salvador Addas-Zanata

Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo

Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil

Abstract

In this note we give a simple computable criteria that assures the existence of hyperbolic horseshoes for certain diffeomorphisms of the torus.

The main advantage of our method is that it is very easy to check numer- ically whether the criteria is satisfied or not.

Key words: twist mappings, topological methods, Nielsen-Thurston theory, Pesin theory

e-mail: sazanata@ime.usp.br

partially supported by CNPq, grant: 301485/03-8

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1 Introduction

In this note, we present a simple criteria, which gives information on the existence of hyperbolic horseshoes for certain diffeomorphisms (the so called twist diffeomorphisms) of the 2-torus. As is well-known, the existence of a horseshoe for a dynamical system implies ”chaotic” behavior, so an important subject in dynamical systems theory is to develop methods that assure the presence of horseshoes for a given system. What we present here is a simple application of some well-known important results to a particular class of diffeomorphisms of the 2-torus, the ones which satisfy a twist condition.

In order to be more precise, let us give some definitions.

Definitions

0) Let T²= IR²/ZZ² be the flat torus and let (φ, I)∈T² denote coordinates in the torus, (bφ,I)b ∈ S¹×IR=(IR/ZZ)×IR denote coordinates in the cylinder, wherebφis defined modulo 1. Let p1 :S¹×IR→IR and p2 :S¹×IR→IR be given by:p1(bφ,I) =b bφand p2(bφ,I) =b Ib(the corresponding projections in the plane are denoted in the same way).

1) LetDif f_t¹⁺(T²) be the set ofC¹⁺(for any >0) torus diffeomorphisms T(φ, I) = (Tφ(φ, I), TI(φ, I)) that are homotopic to the Dehn twist (φ, I) → (φ+I mod1, I mod1) and satisfy∂ITφ≥K >0,whereKis a positive number (this is the so called twist condition).

2) LetD_t¹⁺(S¹×IR) be the set of diffeomorphisms of the cylinderTb(bφ,I)b which are lifts of elements fromDif f_t¹⁺(T²).Clearly, such aTbsatisfies:Tb(bφ,I+b 1) =Tb(bφ,I) + 1.b

Now we are ready to state our result:

Theorem 1 : Given T ∈Dif f_t¹⁺(T²), there exists a number M > 0, which depends only onT and can be easily computed, such that if for some lift Tb of T, there are points bz,wb∈ S¹×IR and natural numbers n > 1, k > 1 such that p₂◦Tbⁿ(z)b −p₂(bz)> M andp₂◦Tb^k(w)b −p₂(w)b <−M, thenT has a hyperbolic periodic point, whose stable and unstable manifolds intersect transversally.

The proof of this result will show thatM can be computed fromT in a very simple way and asTb(bφ,Ib+ 1) =Tb(bφ,I) + 1, one just has to iterate a horizontalb curveIb=const.a sufficiently large number of times in order to see if the curve contains points likebzandwbas in the theorem.

This paper is organized as follows. In the next section we present the state- ments of some results we use. In section 3 we present the proof of our result.

2 Basic tools

Here we recall some topological results for twist mappings essentially due to Le

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the plane. For every pair (s, q), s∈ZZ andq∈IN^∗ we define the following sets (π: IR²→S¹×IR is given byπ(eφ,I) = (ee φ mod1,I)):e

K_{lif t}(s, q) =n

(eφ,I)e ∈IR²: p₁◦Te^q(eφ,I) =e eφ+so and

K(s, q) =π◦Klif t(s, q)

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Then we have the following:

Lemma 1 : For every s ∈ ZZ and q ∈ IN^∗, K(s, q) ⊃ C(s, q), a connected compact set that separates the cylinder.

For alls∈ZZ andq∈IN^∗ we can define the following functions onS¹: µ⁻(bφ) = min{p2(z): z∈K(s, q) andp1(z) =bφ}

µ⁺(bφ) = max{p2(z): z∈K(s, q) andp1(z) =φ}b And we can define similar functions forTb^q(K(s, q)):

ν⁻(bφ) = min{p2(z): z∈Tb^q◦K(s, q) andp1(z) =bφ}

ν⁺(bφ) = max{p2(z): z∈Tb^q◦K(s, q) andp1(z) =bφ}

Lemma 2 : Defining Graph{µ^±}={(bφ, µ^±(bφ)) :φb∈S¹} we have:

Graph{µ⁻} ∪Graph{µ⁺} ⊂C(s, q) So for all bφ∈S¹ we have(bφ, µ^±(bφ))∈C(s, q).

The next lemma is a fundamental result in all this theory:

Lemma 3 : Tb^q(bφ, µ⁻(bφ)) = (bφ, ν⁺(bφ))andTb^q(bφ, µ⁺(bφ)) = (bφ, ν⁻(bφ)).

For proofs of the previous results see Le Calvez [6] and [7].

Now we remember some ideas from [8].

Given a triplet (s, p, q) ∈ ZZ²×IN^∗, if there is no point (eφ,I)e ∈ IR² such that Te^q(eφ,I) = (ee φ+s,Ie+p), it can be proved that the sets Tb^q◦K(s, q) and K(s, q) + (0, p) can be separated by the graph of a continuous function from S¹ to IR, essentially because from all the previous results, either one of the following inequalities must hold:

ν⁻(bφ)−µ⁺(bφ)> p (2) ν⁺(bφ)−µ⁻(bφ)< p (3) for allφb∈S¹,where ν⁺, ν⁻, µ⁺, µ⁻ are associated toK(s, q).

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3 Proofs

The proof of theorem 1 depends on the next lemma, which is a variation of a result of [1]. First we need some definitions:

GivenTb∈D¹⁺_t (S¹×IR) and a liftTe∈D¹⁺_t (IR²), it can be written in the following way,

Te: (

eφ⁰=Teφ(eφ,I)e Ie⁰=TeI(eφ,I)e and for all (eφ,I)e ∈IR² we have the following estimates:

∃a >0, such that

Te_I(eφ,I)e −Ie

< a (4)

∃b >0, such that

∂Teφ

∂eφ

< b (5)

∃K >0, such that ∂Teφ

∂Ie ≥K (twist condition) (6) Now, letM =h

3 +^(2+b)_K i

+a >0.

Lemma 4 : Let Te ∈ D¹⁺_t (IR²) be such that, there exist points z, w ∈ [0,1]² and natural numbers n > 1, k > 1, such that p2 ◦Teⁿ(z)−p2(z) > M and p₂◦Te^k(w)−p₂(w) <−M. Then there are points z⁰, w⁰ ∈ [0,1]² and numbers n⁰, k⁰∈IN^∗ such that

(

Teⁿ⁰(z⁰) =z⁰+ (s, pos)

Te^k⁰(w⁰) =w⁰+ (l, neg) ,for somes, l∈ZZ, pos≥1 andneg≤ −1.

Proof.

The proof of existence ofw⁰ is analogous to the one forz⁰,so we omit it.

By contradiction, suppose that for allx∈[0,1]² (as Te is the lift of a torus mapping homotopic to the Dehn twist, we can restrict ourselves to points in the unit square) andn >0 such thatTeⁿ(x) =x+ (s, m),for some s∈ZZ, we have m <1.

A very natural thing is to look for a point z⁰ as above, in the line segment r=n

(eφ,I)e ∈[0,1]²: eφ=p₁(z) =φe_zo . First, let us define

M ax.H.L(Teⁿ(r)) = sup eI,e^I⁰^∈[0,1]

p₁◦Teⁿ(eφ_z,I)e −p₁◦Teⁿ(eφ_z,Ie⁰)

. (7) It is clear that

_n→∞

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So, for alln >1,∃at least ones∈ZZ,such thateφ_z+s∈p1

Teⁿ(r)

. The hypothesis we want to contradict implies that for alln > 0 andx∈r, such that

p₁◦Teⁿ(x) =eφ_z (mod1), (9) we have:

p₂◦Teⁿ(x)<1 +p₂(x)<2 (10) Asp2◦Teⁿ(z)>h

3 +^(2+b)_K i

+a,∃z1∈rsuch that:

p2◦Teⁿ(z1) = 3 +a and

∀x∈ z₁z⊂r, p₂◦Teⁿ(x)≥3 +a

The reason why such a pointz1exists is the following: Asn >1,∃at least one s ∈ ZZ such that eφ_z+s ∈ p₁

Teⁿ(r)

. Thus, from (9) and (10), Teⁿ(r) must cross the linel given by: l=n

(eφ,3 +a), witheφ∈IRo Also from (9) and (10) we have that:

sup

x,y∈z1z

p₁◦Teⁿ(x)−p₁◦Teⁿ(y) <1 Now letγ_n :J →IR²be the following curve:

γ_n(t) =Teⁿ(eφ_z, t), t∈J= interval whose extremes arep₂(z) andp₂(z₁) (11) It is clear that it satisfies the following inequalities:

p₂◦γ_n(p₂(z))−p₂◦γ_n(p₂(z₁))> ^(2+b)_K sup

t,s∈J

|p1◦γ_n(t)−p₁◦γ_n(s)|<1 Claim 1 : Given a continuous curveγ:J = [α, β]→IR²,with

sup

t,s∈J

|p1◦γ(t)−p1◦γ(s)|<1 (12)

|p2◦γ(β)−p2◦γ(α)|>(2 +b)

K (13)

Then∃s∈ZZ,such thateφ_z+s∈p₁

Te◦γ(J) .

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Proof.

sup

t,s∈J

p₁◦Te◦γ(t)−p₁◦Te◦γ(s) =

= sup

t,s∈J

Teφ◦γ(t)−Teφ◦γ(s) ≥

Teφ◦γ(β)−Teφ◦γ(α) ≥

≥ −b+K.^(2+b)_K = 2 So the claim is proved.

γ_n(t) (see (11)) satisfies the claim hypothesis, by construction. So∃ s∈ZZ such thateφ_z+s∈p1

Te◦γ_n(J)

=p1

Teⁿ⁺¹(z1z) . As inf

t∈Jp2(γ_n(t)) =p2(γ_n(p2(z1))) = 3 +a,from the choice ofa >0 we get that inf

t∈Jp2

Te◦γ_n(t)

>3.

So there existst⁰∈J andz⁰= (eφ_z, t⁰)∈rsuch that:

p1◦Teⁿ⁺¹(z⁰) =eφ_z (mod 1) p2◦Teⁿ⁺¹(z⁰)>3

And this contradicts (9) and (10). So ∃ n⁰ > 0 and z⁰ ∈ r, such that Teⁿ⁰(z⁰) =z⁰+ (s, pos) for somes∈ZZ,withpos >1.

Thus from theorem 1 hypotheses, applying lemma 4, we get that there are pointsz⁰, w⁰∈S¹×[0,1] and numbersn⁰, k⁰∈IN^∗ such that

Tbⁿ⁰(z⁰) =z⁰+ (0, pos)

Tb^k⁰(w⁰) =w⁰+ (0, neg) , for constantspos >1 andneg <−1.

For some s, l ∈ ZZ, z⁰ ∈ K(s, n⁰) and w⁰ ∈ K(l, k⁰), see definition (1) and lemma 1. If we remember lemmas 2 and 3, we get thatν⁺(p₁(z⁰))−µ⁻(p₁(z⁰))≥ pos >1.So, we have 2 possibilities:

i) there exists z^∗∈C(s, n⁰) such thatTbⁿ⁰(z^∗) =z^∗+ (0,1)

ii) as C(s, n⁰) is connected, for allx∈C(s, n⁰), p2◦Tbⁿ⁰(x)−p2(x)>1 ⇒ ν⁻(bφ)−µ⁺(bφ)>1,for allφb∈S¹.

So there exists a simple closed curveγ⊂S¹×IR,which is a graph overS¹, that satisfies:

Tbⁿ⁰(γ) is aboveγ+ (0,1) (14) AsTbis the lift of a torus mapping homotopic to the Dehn twist, expression (14) implies thatTbⁿ⁰(γ+ (0, m)) is aboveγ+ (0,1 +m),for allm∈ZZ. So, for allx∈S¹×IR,

lim inf

n→∞

p2◦Tbⁿ(x)−p2(x)

n ≥ 1

n⁰ >0. (15)

Now we consider w⁰ ∈ K(l, k⁰). In the same way as above, we get that ν⁻(p1(w⁰))−µ⁺(p1(w⁰))≤neg <−1.So, again, we have 2 possibilities:

a) there existsw^∗∈C(l, k⁰) such thatTb^k⁰(w^∗) =w^∗−(0,1).But this clearly

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b) asC(l, k⁰) is connected, for allx∈C(l, k⁰), p2◦Tb^k⁰(x)−p2(x)<−1⇒ ν⁺(bφ)−µ⁻(bφ)<−1,for allbφ∈S¹.

So there exists a simple closed curveα⊂S¹×IR,which is a graph overS¹, that satisfies:

Tb^k⁰(α) is belowα−(0,1) (16) As above, expression (16) implies thatTb^k⁰(α+(0, m)) is belowα+(0, m−1), for allm∈ZZ.So, for allx∈S¹×IR,

lim sup

n→∞

p2◦Tbⁿ(x)−p2(x)

n ≤ −1

k⁰ <0,

which again contradicts (15). Thus possibility ii) above does not happen and we get that there existsz^∗∈C(s, n⁰) such thatTbⁿ⁰(z^∗) =z^∗+ (0,1).From the existence ofz^∗, possibility b) can not happen and so there existsw^∗∈C(l, k⁰) such that Tb^k⁰(w^∗) = w^∗−(0,1). The orbits of z^∗ and w^∗ project to periodic orbits forT,the mapping on the torus. LetOz^∗ andOw^∗ denote these periodic orbits and let Q = Oz^∗ ∪Ow^∗. Now blow-up each x ∈ Q to a circle Sx. Let T²_Q be the compact manifold (with boundary) thereby obtained ; T²_Qis the compactification of T²\Q, where Sx is a boundary component where x was deleted. Now we extend T : T²\Q→T²\Q to T_Q : T²_Q→T²_Q by defining T_Q :S_x →S_x via the derivative; we just have to think ofS_xas the unit circle inT_xT² and define

TQ(v) = DTx(v)

kDTx(v)k, forv∈Sx.

T_Qis continuous on T²_QbecauseT isC¹on T².Letb: T²_Q→T²be the map that collapses eachS_xontox.ThenT◦b=b◦T_Q.This givesh(T_Q)≥h(T) (see [5], page 111). Actually h(T_Q) = h(T), because each fibre b⁻¹(y) is a simple point or anS_x and the entropy ofT on any of these fibres is 0 (the map on the circle induced from any linear map has entropy 0). This construction is due to Bowen (see [2]). In [1] we proved the following theorem:

Theorem 2 : The mapping T_Q : T²_Q→T²_Q is isotopic to a pseudo-Anosov homeomorphism ofT²_Q.

In a certain sense, pseudo-Anosov homeomorphisms are the less ”complex”

mappings in their isotopy classes. One of the many motivations for this impre- cise notion is the following theorem (see [3] and [4] for a proof and for more information on the theory):

Theorem 3 : Let M be a compact, connected oriented surface possibly with boundary, andf, g:M →M be two isotopic homeomorphisms. If g is pseudo- Anosov, then their topological entropies satisfy:h(f)≥h(g)>0.

The above results imply thath(T) =h(TQ)>0.Finally, in order to conclude the proof, we just have to remember the following result due to Katok, see for instance the appendix of [5].

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Lemma 5 : Let f :M → M be a C¹⁺ (for any > 0) diffeomorphism of a closed surface M. If h(f) > 0, then f has a hyperbolic periodic point, whose stable and unstable manifolds intersect transversally.

And the proof of our criteria is complete.

References

[1] Addas-Zanata S. (2002): On the existence of a new type of periodic and quasi-periodic orbits for twist maps of the torus.Nonlinearity15 (5), 1399- 1416

[2] Bowen R. (1978): Entropy and the Fundamental Group.Springer Lec. Notes in Math.668,21-29

[3] Fathi A., Laudenbach F. and Poenaru V. (1979): Travaux de Thurston sur les surfaces.Ast´erisque.66-67

[4] Handel M. and Thurston W. (1985): New proofs of some Results of Nielsen.

Adv. Math.56,173-191

[5] Katok A. and Hasselblatt B. (1995): Introduction to Modern Theory of Dy- namical Systems.Cambridge University Press

[6] Le Calvez P. (1986): Existence d’orbites quasi-p´eriodiques dans les at- tracteurs de Birkhoff. Comm. Math. Phys.106, 383-394

[7] Le Calvez P. (1991): Propriétés Dynamiques des Difféomorphismes de L’Anneau et du Tore.Astérisque204

[8] Le Calvez P. (1995): Construction d’orbites périodiques par perturbation d’un difféomorphisme de l’anneau déviant la verticale. C. R. Acad. Sci.

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