Dynamics of a generic dissipative homeomorphism

In their article [AA13], F. Abdenur and M. Andersson try to identify the generic ergodic properties of continuous maps and homeomorphisms of compact manifolds.

More precisely, they study the behaviour of Birkhoff limits¹ µ^f_x for a generic homeo-morphismf ∈Homeo(X) and almost every pointxforλ. To do this, they define some interesting behaviours of homeomorphisms related to Birkhoff limits, including one they callweird.

Definition 4.1. A homeomorphismf is saidweirdif almost every pointx ∈X (for λ) has a Birkhoff’s limitµ^f_x, and iff istotally singular(i.e. there exists a Borel set with total measure whose image byf is null measure) and does not admit any physical measure.

This definition is supported by their proof, based on the shredding lemma, that a generic homeomorphism is weird. We state an improvement of this lemma, whose main consequence is that a generic homeomorphism has many open attractive sets, all of small measure, and decomposable into a small number of small diameter open sets:

Lemma 4.2(Shredding lemma, F. Abdenur, M. Andersson, [AA13]). For every homeo-morphismf ∈Homeo(X), for allε,δ>0, there exists integers` and`₁,· · ·, `<sub>`</sub>, bigger than

e) We can further assume that each setW_j,icontains a Lyapunov stable periodic point (see Lemma4.4).

Moreover, these properties remain true on a neighbourhood of the homeomorphismg.

We outline the proof of this lemma: by using arguments such as the Oxtoby-Ulam theorem (Theorem3.14) or the concept of uniform grid on the cube Iⁿ, it is possible to shorten a little the arguments of [AA13].

1. We recall that the Birkhofflimitµ^f_x is the limit in the Cesàro sense of the pushforwards of the Dirac measureδ_xby the homeomorphismf.

2. An open set is saidregularif it is equal to the interior of its closure.

3. These two sets are decreasing intersections of compact sets, thus compact and non-empty.

• •

Figure 4.1: Local perturbation for Lemma4.2

Proof of Lemma4.2. Let f ∈ Homeo(X) and δ,ε>0. To begin with, we endow X with a collection of “cubes”: for N large enough, the images of the cubes associated to the the grid E¹_N on the cube Iⁿ (see page42) by the map given by Oxtoby-Ulam theorem (Theorem3.14) give us a collection of cubes (C_i) such that their union has full measure, their boundaries have null measure, they all have the same measure and their diameters are smaller thanδ. To each cube C_i we associate its centre e_i. These properties assert that this collection of cubes behaves like the collection of cubes on Iⁿ endowed with Lebesgue measure. Thus, in the sequel, we will only treat the case of Iⁿ endowed with Leb.

We define a finite mapσ:{e_i} → {e_i}such thatf(e_i)∈C_σ(ei). Making a perturbation if necessary, we can suppose thatσ does not contain any periodic orbit of length smaller than 1/ε (simply because if N is large enough, then for all e_i there exists at least 1/ε other pointse_j such thatd(e_i, e_j)<δ). We then use Corollary3.4to move the pointsf(e_i) closer to the pointse_j, in other words we build a homeomorphismg₁which isδ-close to f and such that max_i

min_jd(g₁(e_i), e_j)

<min_idiam(C_i)/10.

We then set g₂ =g₁◦h, where his a homeomorphism close to the identity whose restriction to every cube is a huge contraction (see Figure4.1): such a contraction can be easily expressed on the cube [−1,1]ⁿ: letαbe a “big” number (determined by min(ε,ε⁰)) and

φ_α(x) =

( kxk^α_∞x ifkxk_∞≤1 x otherwise.

We then build easilyhby composing such contractions on each cube, each of them being obtained by conjugating by a translation and a homothety.

The “physical” dynamics of the map g₂ is then close to that of the finite map σ which mapse_i on the unique centre e_j such thatg(e_i)∈C_j. In particular, the periodic sets W_j,1,· · ·,W_j,`j are obtained as neighbourhoods of a given periodic orbit ofσ, and the corresponding basins of attraction U_jare unions of subsets of the cubes⁴whose centres have some iterates byσ which fall in this periodic orbit. We then easily check that these sets satisfy the conclusions of the lemma, apart from the point(ii).

Thus, we have to prove that the basins of attraction can be supposed to have small measure. To do that, we consider the cubes of order M =kN withk >1/ε, and partition this set of cubes intokⁿsets of cubes in the way defined by the fact that these grids are strongly self similar (see page42). We then use this self similarity to perturbf such that σstabilizes each one of these subgrids, and apply the same technique as previously.

4. More precisely, for every cube C_j, the sets U_jcontain{x∈C|d(x∂C_j)>diam(C)(1−ε)}.

Definition 4.3. For a homeomorphismf ∈Homeo(X), we denote by A₀the set of Lya-punov stableperiodic points off,i.e. the set of periodic pointsxsuch that for allδ>0, there existsη>0 such that ifd(x, y)<η, thend(f^m(x), f^m(y))<δfor allm∈N.

For two compact sets K and K⁰, by K⁰⊂⊂K we mean that there exists an open set O such that K⁰ ⊂O⊂K. In the sequel the set K will be calledstrictly periodicif there exists an integeri >0 such thatfⁱ(K)⊂⊂K.

The following lemma ensures that for a generic homeomorphism, every set W_j,i of the shredding lemma contains at least one (in fact an infinite number of) Lyapunov stable periodic point.

Lemma 4.4. For a generic homeomorphismf ∈Homeo(X), for every strictly periodic topo-logical ballO(i.e. there existsi >0such thatfⁱ(O)⊂⊂O), there exists a Lyapunov stable periodic pointx∈O.

Proof of Lemma4.4. We begin by choosing a countable basis of closed sets of X: for example we can takeKNthe set of unions of the closures of the cubes of order N. We also denote by B the set of all closed topological balls of X. We define Uk,ε,N as the set of homeomorphisms such that each large enough strictly periodic ball contains a smaller strictly periodic ball with the same period⁵:

Uk,ε,N=

Then for every k,ε,N, it is straightforward that the set Uk,ε,N is an open subset of Homeo(X). To show that it is dense it suffices to apply Brouwer’s theorem to each K such thatfⁱ(K)⊂⊂K and to make the obtained periodic point attractive.

We now prove that every f ∈ T

k,ε,NUk,ε,N satisfies the conclusions of the lemma.

First of all, remark that for every topological ball K with non-empty interior which is strictly i-periodic, there exits N ∈ Nand a smaller topological ball ˜K⊂K which is strictly i periodic such that ˜K∈KN. It implies that iff belongs to the G_δ dense set T

k,ε,NUk,ε,N, then for every topological ball K with non-empty interior which is strictly i-periodic, there exits N∈Nand a topological ball K⁰⊂K˜ ⊂K which is strictlyiperiodic and at least twice smaller. Taking the intersection of such balls, we obtain a periodic point with periodiwhich is Lyapunov stable by construction.

The shredding lemma tells us a lot about the dynamics of a generic homeomor-phism, which becomes quite clear: there are many attractors whose basins of attraction are small and attract almost all the manifold X. Moreover there is convergence of the attractive sets of the shredding lemma to the closure of the set of Lyapunov stable peri-odic points off.

Corollary 4.5. Letf ∈Homeo(X)verifying the conclusions of the shredding lemma for all ε=ε⁰>0, andW_j,i,εbe the corresponding open sets. Such homeomorphisms form aG_δdense subset ofHomeo(X). Then the sets

A_ε=[

j,i

W_j,i,ε

5. For a compact set K, diam_int(K) denotes the diameter of the biggest euclidean ball included in K.

converge for Hausdorffdistance whenεtends to0to a closed set which coincides generically with the setA₀(see Definition4.3).

Moreover, generically, the setA₀is a Cantor set (that is, it is compact, without any iso-lated point and totally disconnected) whose Hausdorffdimension⁶is 0.

Remark4.6. Thus, for a generic homeomorphismf ∈Homeo(X), theω-limit set of al-most every pointx∈X is included in A₀.

Proof of Corollary4.5. Letf verifying the hypothesis of the corollary. We want to show that the sets A_εtend to A₀ for Hausdorffdistance whenεgoes to 0. This is equivalent to show that for all δ>0, there existsε₀>0 such that for allε<ε₀, A₀ ⊂B(A_ε,δ) and A_ε⊂B(A₀,δ) (where B(A,δ) denotes the set of points of X whose distance to A is smaller thanδ). Subsequently we will denote by U_j,ε and by W_j,i,ε the open sets given by the shredding lemma for the parametersε=ε⁰.

Let δ>0. We start by taking x∈X whose orbit is periodic with periodpand Lya-punov stable. Then there existsη>0 such that ifd(x, y)<η, thend(f^m(x), f^m(y))<δ/2 for all m ∈ N; we note O = B(x,η). As λ(O) > 0 there exists ε₀ > 0 such that for all ε∈]0,ε₀[, there existsj∈Nsuch that the intersection between O and U_j,εis non-empty.

Let y be an element of this intersection. By compactness, there exists a subsequence of (f^pm(y))_m∈N which tends to a point x₀; moreover f^m(y) ∈ S

iW_j,i eventually and d(f^pm(x), f^pm(y)) < δ/2, thus at the limit m → +∞, d(x, x₀) < δ/2. We deduce that x ∈ BS

j,iW_j,i,ε0,δ/2

for all ε₀ small enough. Since A₀ is compact, it is covered by a finite number of balls of radiusδ/2 centred at some pointsx_i whose orbits attract non-empty open sets. Takingε⁰₀the minimum of all the ε₀associated to thex_i, the inclusion A₀⊂B(A_ε,δ) occurs for allε<ε⁰₀.

Conversely, letδ>0,ε<δand focus on the set W_j,i,ε. By Lemma4.4, we can suppose that there existsx∈W_j,i,εwhose orbit is periodic and Lyapunov stable. Thusx∈A₀and since the diameter of W_j,i,εis smaller thanδ, W_j,i,ε⊂B(A₀,δ).

We now prove that the set A₀has no isolated point. Letx ∈A₀ andδ>0, we want to find another pointy ∈A₀ such thatd(x, y)<δ. If x∈A₀\A₀then it is trivial thatx is an accumulation point of A₀; thus we suppose thatx∈A₀. Asxis a Lyapunov-stable periodic point, there exists a neighbourhood O ofx whose diameter is smaller than δ and which is periodic (i.e. there existst >0 such thatf^t(0)⊂O). Forεsmall enough, the open set O meets at least two sets U_j,εand U_j⁰_,εof the shredding lemma. Thus, applying the shredding lemma, we deduce that O contains at least two different strictly periodic sets W_j,i,εand W_j⁰_,i⁰_,ε; by Lemma4.4they both contain a Lyapunov-stable periodic point, and by construction their distance toxis smaller thanδ.

Finally we prove that generically, the set A₀ has 0 Hausdorffdimension. Lets >0.

We consider the set of homeomorphisms verifying the conclusions of the shredding lemma for ε⁰ such that P

j`_jε^0s < 1. This equality implies that for all ε > 0, the s-Hausdorffmeasure of the setT

δ∈]0,ε[A_δis smaller than 1. As we have A₀=[

k≥0

\

δ∈]0,1/k[

A_δ,

the set A₀is a countable union of sets of Hausdorffdimension smaller thansfor every s >0. Thus, A₀has zero Hausdorffdimension; this also proves that A₀is perfect.

6. And better, if we are given a countable family (λ_m)_m∈Nof good measures, generically the Hausdorff dimension of this set with respect to these measures is zero.

Remark4.7. We can also prove that for everyτ>0, the set of periodic points with period τof a generic homeomorphism is either empty, either a Cantor set with zero Hausdorff dimension.

From the shredding lemma, we can easily deduce that the sequence of pushforwards of the uniform measure on an open set U is Cauchy. Thus, it converges to a measureµ^f_U, which is supported by A₀and is atomless (because theµ^f_U-measure of each setS

iW_j,i,ε is betweenεand 2ε).

Corollary 4.8. For a generic homeomorphismf ∈Homeo(X), for every open subsetUofX, the measureµ^f_Uis well defined, atomless and is supported byA₀.