Posted by: matheuscmss | March 18, 2013

Ergodicity of conservative diffeomorphisms (II)

Sylvain Crovisier gave on February 22, 2013, a second talk — this time at Eliasson-Yoccoz seminar in Jussieu — about his joint work with Artur Avila and Amie Wilkinson that we started to discuss a few weeks ago. In fact, last time we saw that two of the main results of Avila-Crovisier-Wilkinson are:

Theorem 1 (A. Avila, S. Crovisier and A. Wilkinson) There exists {\mathcal{G}\subset\textrm{Diff}^{\,1}_v(M)} a residual (i.e., {G_{\delta}}-dense) subset such that for any {f\in\mathcal{G}}:

  • (ZE) either all Lyapunov exponents {\lambda_i(x)} of {v}-a.e. {x\in M} vanish,
  • (NUA) or {f} is non-uniformly Anosov in the sense that
    • {f} has a (global) dominated splitting, i.e., there is a decomposition {TM=E\oplus F} into {Df}-invariant subbundles such that {F} dominates {E}, that is, there exists {N\geq 1} with {\|Df^N(u)\|\leq (1/2)\|Df^N(v)\|} for any {u\in E}, {v\in F} unitary vectors (“the largest expansion along {E} is dominated by the weakest contraction in {F}, but, a priori, neither {E} is assumed to be contracted nor {F} is assumed to be expanded”).
    • for {v}-a.e. {x\in M}, the fibers of {E_x} and {F_x} of the dominated splitting coincide with the stable and unstable Oseledets subspaces, i.e., {E_x=\mathcal{E}_x^s} and {F_x=\mathcal{E}_x^u},

    and {v} is ergodic.

Theorem 2 (A. Avila, S. Crovisier, A. Wilkinson) For {r>1}, the set of ergodic diffeomorphisms in {\textrm{Diff}^{\,r}_v(M)} contains a {C^1}-open, {C^1}-dense subset of the set {PH^r_v(M)} of partially hyperbolic volume-preserving {C^r}-diffeomorphisms.

Furthermore, we saw a sketch of proof of Theorem 1 based on Sylvain’s talk at LAGA.

Today we’ll focus exclusively on the proof of Theorem 2 based on Sylvain’s talk at Jussieu (assuming, of course, that the reader is familiar with our previous post on this subject).

1. Stable transitivity

Before attacking the problem of stable ergodicity (in {C^1}-topology) in the statement of Theorem 2, let us discuss the issue of getting the weaker property of stable transitivity.

In this direction, we have the following result:

Theorem 3 There exists a {C^1}-open, {C^1}-dense subset of {PH^r_v(M)} consisting of diffeomorphisms such that the orbits of {v}-a.e. {x\in M} is dense.

As it was explained by Sylvain, this theorem follows from the works of D. Dolgopyat and A. Wilkinson, and K. Burns, D. Dolgopyat, and Y. Pesin.

Indeed, by the results of Dolgopyat-Wilkinson, there exists a {C^1}-open, {C^1}-dense subset of {PH^r_v(M)} consisting of accessible partially hyperbolic diffeomorphisms, that is, partially hyperbolic diffeormorphisms {f} such that any pair of points {x,y\in M} can be connected by a finite sequence {x=x_0, x_1, \dots, x_k=y} of points whose stable and unstable manifolds are “related” in the sense that for any {i=0,\dots, k-1}, {x_{i}} and {x_{i+1}} belong to the same stable or unstable leaf.

Intuitively, it is not hard to convince oneself that accessibility has something to do with transitivity/ergodicity: for example, the direct product {f=A\times \textrm{id}} on {M=N\times S} of an Anosov diffeomorphism {A} on {N} by the identity map on {S} is a partially hyperbolic diffeomorphism that is not transitive (nor ergodic) in part because the stable and unstable manifolds of {f} are “confined” to the submanifolds {M\times\{p\}}, {p\in S} (and this is the “opposite” of the accessibility property).

Using the accessibility property, one can obtain “stable transitivity” (in the sense of Theorem 3) from the following idea of M. Brin (further developed by Burns-Dolgopyat-Pesin here). In order to show that the orbit of {v}-a.e. point of {M} is dense, it suffices to show that, given two open sets {U} and {V}, the orbit of {v}-a.e. {x\in U} passes through {V}. Fix {p\in U} and {q\in V} arbitrarily. By the accessibility property, we can join {p} and {q} with a finite path {p=x_0, \dots, x_k=q} such that {x_i} and {x_{i+1}} belong to the same stable of unstable leaf. By using the stable and unstable foliations, we can “propagate” {U} and {V} to construct neighborhoods {U_i} of {x_i} saturated by pieces of stable or unstable leaves.

We proceed by induction, that is, we will show that the orbits of {v}-a.e. point of {U_i} passes through {U_{i+1}}. By Poincaré’s recurrence theorem, {v}-a.e. point is recurrent. Consider the subset {W_{i+1}\subset U_{i+1}} of recurrent points. Since {f} is {C^2}, we can use the absolute continuity of the stable and unstable foliations to see that there exists a subset {W_i\subset U_i} such that {v(U_i-W_i)=0} and each point {p_i} of {W_i} belongs to the stable or unstable leaf of a point {p_{i+1}} of {W_{i+1}}. Since stable and unstable leaves are uniformly contracted to the future or to the past, we deduce that future or past iterates of {p_i} accumulate the respective future or past iterates of {p_{i+1}}. Because {p_{i+1}\in W_{i+1}} is recurrent, it follows that the orbit of each {p_i\in W_i} visits {U_{i+1}}. Of course, this completes the proof of Theorem 3: starting from {U_0=U}, we know that the orbit of {v}-a.e. {x\in U_0} passes through {U_1}, …, the orbit of {v}-a.e. {x\in U_{k-1}} passes through {U_k=V}, and, thus, the orbit of {v}-a.e. {x\in U} passes through {V}.

After this quick discussion of stable transitivity, let us study a mechanism of stable ergodicity inspired by this article of F. Rodriguez-Hertz, J. Rodriguez-Hertz, A. Tahzibi and R. Ures.

2. {(\textrm{RH})^2}-T-U ergodicity criterion

In their article, F. Rodriguez-Hertz, J. Rodriguez-Hertz, A. Tahzibi and R. Ures combine stable transitivity with a “generalized Hopf argument” to prove the following result.

Let {f} be a {C^2} diffeomorphism with a hyperbolic periodic point {O}. Denote by

\displaystyle \mathcal{H}^s(O):=\{x\in M: W^u(x)\sqcap W^s(O)\neq\emptyset\}

and

\displaystyle \mathcal{H}^u(O):=\{x\in M: W^s(x)\sqcap W^u(O)\neq\emptyset\}

where {W^s(x)} and {W^u(x)} denote the Pesin stable and unstable manifolds of {x}, and the symbol {\sqcap} denotes transverse intersection.

Theorem 4 (Rodriguez-Hertz–Rodriguez-Hertz–Tahzibi–Ures) Suppose that {v(\mathcal{H}^s(O))>0} and {v(\mathcal{H}^u(O))>0}. Then:

  • the sets {\mathcal{H}^s(O)}, {\mathcal{H}^u(O)} and {\mathcal{H}(O):=\mathcal{H}^s(O)\cap \mathcal{H}^s(O)} coincide modulo subsets of {v}-measure zero, and
  • {v|_{\mathcal{H}(O)}} is ergodic.

From this result, the idea to approach Theorem 2 is the following. Given {f\in PH_v^r(M)} with a splitting {E^{ss}\oplus E^c\oplus E^{uu}} into (strong/uniform) stable subbundle {E^{ss}}, central subbundle {E^c} and (strong/uniform) unstable subbundle {E^{uu}}, we have from Theorem 1 that there are {f'} {C^1}-close to {f} ergodic non-uniformly Anosov diffeomorphisms. By some semicontinuity arguments (see Section 2 of the previous post), it is possible to check that, for all {g\in PH_v^r(M)} {C^1}-close to {f'}:

  • {v(NUH(g))>0},
  • there is a dominated splitting {TM=E\oplus F} coming from {NUH(f')},
  • there exists a hyperbolic periodic point {O} such that {\textrm{dim}E^s(O)=\textrm{dim}(E)}, and
  • for {v}-a.e. {x}, the eventual zero Lyapunov exponents (if they exist) are all contained in {E} or {F}, i.e., {\textrm{dim}\mathcal{E}^s(x)\geq\textrm{dim}E^s(O)} or {\textrm{dim}\mathcal{E}^u(x)\geq\textrm{dim}E^u(O)}.

At this point, we observe that, by Theorem 4, if we can show that {v(\mathcal{H}^s(O)\cup\mathcal{H}^u(O))=1}, then we deduce that the dynamics of all {g\in PH_v^r(M)} {C^1}-close to {f'} is ergodic.

In other words, the proof of Theorem 2 is reduced to the verification of the equality

\displaystyle v(\mathcal{H}^s(O)\cup\mathcal{H}^u(O))=1.

Here, a naive strategy to check this equality goes as follows. By Theorem 3, we know that the orbit of {v}-a.e. {x\in M} is dense. In particular, the orbit of {x} passes arbitrarily close to {O}, and we could expect that {W^*(x)} intersects transversely {W^{**}(O)} when {\textrm{dim}(\mathcal{E}^*(x))\geq \textrm{dim}(E^*(O))} (here {E^{**}(O)} denotes the complementary invariant subbundle to {E^*(O)}, {*= s} or {u}), i.e., {x\in \mathcal{H}^s(O)\cup\mathcal{H}^u(O)}. However, it is known that Pesin stable/unstable manifolds {W^*(x)} depend measurably on {x}, and, thus, they might get too small to intersect {W^{**}(O)} (so that we can’t conclude that {x\in \mathcal{H}^s(O)\cup\mathcal{H}^u(O)})!

In order to overcome this difficulty, we start by observing that, in our context of partially hyperbolic diffeomorphisms, Pesin invariant manifolds can’t be uniformly small in all directions! More precisely, since {g\in PH_v^r} has a decomposition {E^{ss}\oplus E^c\oplus E^{uu}}, we have that {\mathcal{E}^*(x)} contains {E^{ss}} or {E^{uu}}, and, a fortiori, {W^*(x)} contains {W^{ss}(x)} or {W^{uu}(x)}.

Evidently, this is still not enough to conclude that {x\in \mathcal{H}^s(O)\cup\mathcal{H}^u(O)} because the dimensions of {E^{ss}(x)} and/or {E^{uu}(x)} might be “small” compared to {\mathcal{E}^*(x)} (and in this case there is no hope for transverse intersections).

On the other hand, this hints at the following strategy. After the works of Smale, we know that the “measurable homoclinic class” {\mathcal{H}(O)} of the hyperbolic periodic point {O} usually contains horseshoes {\Lambda}. Moreover, as it was first observed by Christian Bonatti and Lorenzo Diaz, sometimes the horseshoes {\Lambda} might work as blenders (in Bonatti-Diaz terminology), that is, the strong unstable manifolds {W^{uu}(y)} of points {y\in\Lambda} collectively behave like a {\textrm{dim}(E^c\oplus E^{uu})}-dimensional submanifold in the sense that any {\textrm{dim}(E^{ss})}-dimensional submanifold {W} {C^1}-close to a strong stable leave {W^{ss}(z)}, {z\in\Lambda}, crosses some {W^{uu}(y)}. In other terms, a blender {\Lambda} of unstable dimension {d^u=\textrm{dim}(E^c\oplus E^{uu})} is a horseshoe that is “fat” when looked along the strong stable direction {E^{ss}}. The formal definition of a blender goes as follows:

Definition 5 A horseshoe {\Lambda} (i.e., a compact, totally disconnected, locally maximal, hyperbolic set) whose tangent bundle dynamics is {T_{\Lambda}M=E^s\oplus E^u = E^{ss}\oplus E^c\oplus E^{uu}} is a blender of unstable dimension {d^u=\textrm{dim}(E^c\oplus E^{uu})} if there exists a constant {L>0} and a point {p\in\Lambda} such that, if {\gamma} is a {\textrm{dim}(E^{ss})}-dimensional disk {C^1}-close to the local strong stable manifold {W_{L}^{ss}(p)} of {p} of size {L>0}, then {\gamma} intersects {W^u(y)} for some {y\in\Lambda}. (A blender of stable dimension {d^s} is defined in a similar way)

Of course, this phenomenon reminds Marstrand’s theorem that the projections of fat horseshoes in certain directions might contain positive measure sets, and, in fact, some versions of Marstrand’s theorem and the construction of blenders appear in this recent work of C. G. (Gugu) Moreira and W. Silva.

In any event, by pursuing the ideas described in the previous paragraph, Avila-Crovisier-Wilkinson further reduce the proof of their Theorem 2 to the following statement:

Theorem 6 Given a {C^r}-diffeomorphism {f} ({r>1}) of a compact manifold {M} such that {v(NUH(f))>0}, there are {C^r}-diffeomorphisms {g} {C^1}-close to {f} with a blender of unstable dimension {d^u=\textrm{dim}(M)-1} and stable dimension {d^s=\textrm{dim}(M)-1}.

In next (and last) section, we will complete the proof of Theorem 2 by sketching the proof of Theorem 6.

3. Construction of blenders

Let {v_0} be an ergodic component of {v|_{NUH(f)}}. By definition, the (ergodic) probability measure {v_0} is hyperbolic (in Pesin’s sense), i.e., its Lyapunov exponents {\chi_i(v_0)} are all non-zero. Furthermore, since {f} is {C^2}, Pesin’s formula says that

\displaystyle h_{v_0}(f) = \sum_i \max(\chi_i(v_0),0) = \sum_j \max(-\chi_j(v_0),0)

By some results of A. Katok, given {\varepsilon>0}, one can find inside {NUH(f)} a horseshoe {\Lambda} such that:

  • its topological entropy is {\varepsilon}-close to the metric entropy of {v_0} in the sense that {h_{top}(\Lambda)>h_{v_0}(f)-\varepsilon}, and
  • all invariant measures supported on {\Lambda} have Lyapunov exponents {\varepsilon}-close to {\chi_i(v_0)}.

By the works of F. Ledrappier and L.-S. Young, we can “convert” the informations in the two items above into quantitative information about the fractal dimensions of {\Lambda}: very roughly speaking, denoting by {\mu} the maximal entropy measure of {\Lambda} (i.e., {h_{\mu}(f)=h_{top}(\Lambda)}), Ledrappier and Young associated “dimensions” {D_1,\dots, D_d\in [0,1]} ({d=\textrm{dim}(M)}) such that

\displaystyle h_{\mu}(f)=\sum_i D_i\max(\chi_i(\mu),0) = \sum_j D_j\max(-\chi_j(\mu),0)

In our context, we know that {h_{\mu}(f)} is close to {h_{v_0}(f)}, the Lyapunov exponents {\chi_i(\mu)} are close to {\chi_i(v_0)}, so that, by Pesin’s formula, we deduce that the Ledrappier-Young “dimensions” {D_i} are close to {1}, and, as it is explained in Ledrappier-Young’s works, this means that {\Lambda} has large fractal (e.g., Hausdorff) dimension.

Once they dispose of a horseshoe with large fractal dimension, Avila-Crovisier-Wilkinson concentrate their efforts in showing the following theorem:

Theorem 7 Let {f} be a {C^2} diffeomorphism with a horseshoe with “large fractal dimension”. Then, there are {g} {C^2}-diffeomorphisms {C^1}-close to {f} with blenders (of stable and unstable dimensions {d-1}).

Of course, Theorem 6 is a consequence of this theorem.

As Sylvain explained to us, the inspiration for this theorem is the following recent result of C. G. Moreira and W. Silva:

Theorem 8 (Moreira-Silva) Let {f} be a {C^{\infty}}-diffeomorphism with a horseshoe {\Lambda} with a decomposition {E^{ss}\oplus E^c\oplus E^{uu}} with {\textrm{dim}(E^c)=1}. Then, there are {g} {C^{\infty}}-close to {f} with a blender of unstable dimension {d^u=\textrm{dim}(E^{uu})+1}.

The attentive reader noticed the following differences between Theorem 7 and Moreira-Silva’s theorem. In the former, there is no assumption on the dimension of {E^c}, but, in compensation, in the latter one has the stronger conclusion that blenders can be found via {C^{\infty}}-small perturbations.

In fact, Sylvain told that Theorem 7 might also be true with {C^{\infty}}-small perturbations. Indeed, Avila-Crovisier-Wilkinson show Theorem 7 in two steps:

  • Step 1: given {\varepsilon>0}, there are {\Lambda'\subset\Lambda} a subhorseshoe with topological entropy {\varepsilon}-close to {h_{\mu}(f)} and {g} a {\varepsilon}-small {C^1}-perturbation of {f} such that the continuation {\Lambda_g'} of {\Lambda'} is affine, i.e., {\Lambda_g'} has a Markov partition {R_n(g)}, {n=1,\dots, N}, such that, in adequate coordinates {x}, the return maps of {g} to {R_n(g)} have the form {x\mapsto A_n(x)+b_n}, where {A_n} is a matrix and {b_n} is a constant vector (for each {n}).
  • Step 2: if {\Lambda} is an affine horseshoe of {f} with “large fractal dimension”, then there are {C^{\infty}}-perturbations {g} of {f} such that the continuation {\Lambda_g} contains blenders (of stable and unstable dimensions {d-1}).

Here, they use {C^1}-perturbations to place themselves into the favorable situation of affine horseshoes, but, in principle, there is no reason to believe that Step 1 is unavoidable.

Closing his talk, Sylvain said a few words about Steps 1 and 2.

Following him, let us start with Step 2. For the sake of simplicity, let us assume that {M} is a {3}-dimensional manifold, so that our affine horseshoe {\Lambda} has a invariant splitting {T_{\Lambda}M=E^{ss}\oplus E^c\oplus E^{uu}} into three 1-dimensional subbundles. In order to get a blender with unstable dimension 2 out of {\Lambda}, we can “forget” about the {E^{uu}}-direction, that is, we will pretend that {\Lambda} is an affine horseshoe on a {2}-dimensional submanifold, say {\Lambda} is an affine horseshoe living in a square {[0,1]^2\subset\mathbb{R}^2}, where the horizontal direction {\mathbb{R}(1,0)} is {E^{c}} and the vertical direction {\mathbb{R}(0,1)} is {E^{ss}}. In this setting, a blender is a horseshoe in {\mathbb{R}^2} intersecting all vertical lines {\{x\}\times[0,1]}, that is, a horseshoe whose vertical projection contains the interval {[0,1]}. Note that a projection doesn’t increase the Hausdorff dimension, it is clear that such blenders have “large fractal dimension” (e.g. {1+\varepsilon} for some {\varepsilon>0}). Anyhow, the construction of a horseshoe whose vertical projection contains {[0,1]} is not difficult: the basic idea is that this last property is satisfied if the vertical projections of the Markov rectangles {R_n} defining the horseshoe cover {[0,1]}.

In summary, the verification of Step 2 amounts to show that given an affine horseshoe {\Lambda} with large fractal dimension, we can perturb (in {C^{\infty}} topology) the dynamics so that the projections along {E^{ss}} of Markov rectangles defining {\Lambda} overlap considerably. Of course, since {\Lambda} has large fractal dimension, the Markov rectangles have large size. Thus, if we could perturb independently these Markov rectangles, then the desired overlap property would follow. Of course, it is not possible to perturb independently nearby Markov rectangles, but an ingenious probabilistic argument of Moreira-Yoccoz shows that there exists a large (almost full) probability that a random perturbation of the Markov rectangles will move “independently” Markov rectangles reasonably far away from each other.

Finally, let us quickly discuss Step 1. The basic idea is very simple. As we mentioned above, the Lyapunov exponents of all measures supported on {\Lambda} are {\varepsilon}-close to constants (namely, {\chi_i(v_0)}), that is, we are almost affine. Using this information, one selects a tiny grid of our space, and one linearizes (after {C^1}-perturbation) the dynamics inside the elements of the grid. Of course, there is a boundary effect to control (in order to ensure that the dynamics at the boundary of the grid is well-defined) and, in this direction, Avila-Crovisier-Wilkinson use a “reversed doubling property” of the maximal entropy measure {\mu} of {\Lambda} (saying that the ratio of concentric balls are controlled in terms of the ratio of their radii: {\mu(B(x,r\delta))\leq (1/2) \mu(B(x,r))} for some {\delta>0}) to show that this boundary effect is negligible.

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