Posted by: matheuscmss | September 30, 2012

Homoclinic/heteroclinic bifurcations: slightly fat horseshoes B

In what follows, we will consider the same setting of the previous post, and we will discuss a recent improvement obtained in collaboration with Jacob Palis and Jean-Christophe Yoccoz (on a work still in progress). In particular, all statements below concern the dynamics of {\Lambda_g} where {g=g_t} and {t} is a strongly regular parameter in the sense of the article of J. Palis and J.-C. Yoccoz: in other words, in the sequel, we don’t have to exclude further parameters in order to get our (slightly improved) statements, and, indeed, our arguments will be based on soft analysis.

In this post we will prove the following result:

Theorem 1 (C. M., J. Palis and J.-C.Yoccoz) The Hausdorff dimension of the stable and unstable sets {W^s(\Lambda_g)} and {W^u(\Lambda_g)} of the non-uniformly hyperbolic horseshoe {\Lambda_g} is strictly smaller than {2}.

Logically, this result slightly improves Theorem 2 (of J. Palis and J.-C. Yoccoz) in the previous post saying that the {2}-dimensional Lebesgue measures of {W^s(\Lambda_g)} and {W^u(\Lambda_g)} are zero (because any subset of the compact {2}-dimensional manifold {M} with Hausdorff dimension strictly smaller than {2} has zero {2}-dimensional Lebesgue measure).

The plan for the rest of this post (the last one of this series) is the following: in the next section we will prove Theorem 1, and in the final section we will make some comments on further results obtained in collaboration with Jacob and Jean-Christophe.

1. Hausdorff dimension of the stable sets of non-uniformly hyperbolic horseshoes

Let’s start by nicely decomposing the stable set {W^s(\Lambda_g)}. Using the notations of of the previous post of this series, we can write

\displaystyle W^s(\Lambda_g):=\bigcup\limits_{n\geq 0} g^{-n}(W^s(\Lambda_g,\widehat{R})\cap R)

where {W^s(\Lambda_g,\widehat{R}):=\bigcap\limits_{n\geq 0} g^{-n}(\widehat{R})}.

Since {g} is a diffeomorphism, it follows from the basic properties of the Hausdorff dimension (see item (e) of Proposition 3 of this post here) that

\displaystyle \textrm{HD}(W^s(\Lambda_g))=\textrm{HD}(W^s(\Lambda_g,\widehat{R})\cap R)

Now, we separate {W^s(\Lambda_g,\widehat{R})\cap R} into its good (non-uniformly hyperbolic) part and its exceptional part as follows:

\displaystyle W^s(\Lambda_g,\widehat{R})\cap R:=\bigcup\limits_{n\geq 0} (W^s(\Lambda_g,\widehat{R})\cap R\cap g^{-n}(\widetilde{\mathcal{R}}_+^{\infty})) \cup\mathcal{E}^+

In other words, the good part of {W^s(\Lambda_g,\widehat{R})\cap R} consists of points passing by the nice set {\widetilde{\mathcal{R}}_+^{\infty}} of stable curves and the exceptional set {\mathcal{E}^+} is, by definition, the complement of the good part.

The set {\widetilde{\mathcal{R}}_+^{\infty}} is the “nice” (non-uniformly hyperbolic) part of the dynamics and hence it is not surprising that J. Palis and J.-C. Yoccoz showed in Section 10 of their work that {\widetilde{\mathcal{R}}_+^{\infty}} has Hausdorff dimension {1+d_s} where {d_s} is close to the stable dimension {d_s^0} of the initial horseshoe {K}.

Thus, the proof of Theorem 1 is reduced (cf. item (b) of Proposition 3 of this post here) to show that {\textrm{HD}(\mathcal{E}^+)<2}.

Now, we follow the discussion of Section 11.7 of Palis-Yoccoz article to decompose {\mathcal{E}^+} by looking at successive passages through parabolic cores of strips. More concretely, given an element {(P,Q,n)\in\mathcal{R}}, we define the parabolic core {c(P)} of {P} as

\displaystyle c(P)=\{p\in W^s(\Lambda_g,\widehat{R}): p\in P \textrm{ but }p\notin P' \textrm{ for all }P' \textrm{ \emph{child} of }P\}

Here, a child {P'} of {P} is a {(P',Q',n')\in\mathcal{R}} such that {P'\subset P} but there is no {P'\subset P''\subset P} with {(P'',Q'',n'')\in\mathcal{R}}. The geometry of a parabolic core {c(P_k)} of {(P_k,Q_k,n_k)\in\mathcal{R}} is depicted below:

Fig. 1. The parabolic core {c(P_k)} of {P_k} belongs to the grey region inside {P_k}.

By checking the definitions (of good part and exceptional set), it is not hard to convince oneself that {\mathcal{E}^+} can be decomposed as

\displaystyle \mathcal{E}^+=\bigcup\limits_{(P_0,\dots,P_k) \textrm{ admissible}}\mathcal{E}^+(P_0,\dots,P_k)

where {(P_0,Q_0,n_0),\dots,(P_k,Q_k,n_k)\in\mathcal{R}} and the sets {\mathcal{E}^+(P_0,\dots,P_k)} are inductively defined as {\mathcal{E}^+(P_0):=c(P_0)}, {\mathcal{E}^+(P_0,P_1)=\{z\in\mathcal{E}^+(P_0): g^{n_0}(G(z))\in c(P_1)\}}, {\dots} (cf. Equations (11.57) to (11.63) of Palis-Yoccoz paper). Here, we say that {(P_0,\dots,P_k)} is admissible if {\mathcal{E}^+(P_0,\dots,P_k)\neq\emptyset}.

As the picture above indicates, the fact that the points of {\mathcal{E}^+(P_0,\dots,P_k)\neq\emptyset} passes by successive parabolic cores imposes strong conditions over the elements {(P_i,Q_i,n_i)}: for instance, the parabolic core {c(P_i)} of any {P_i} is non-empty, the horizontal bands {Q_i} are always critical and, because we’re dealing {g=g_t} where {t} is a strongly {\beta}-regular parameter, the following estimate holds:

Lemma 1 (Lemma 24 of Palis-Yoccoz article) Suppose that {\mathcal{E}^+(P_0,\dots,P_{j+1})\neq\emptyset}, where {j\geq 1}. Then,

\displaystyle \max(|P_{j+1}|,|Q_{j+1}|)\leq C|Q_j|^{\widetilde{\beta}}

where {\widetilde{\beta}=\beta(1-\eta)(1+\tau)^{-1}>1} and {\beta >1} and {\eta\ll\tau\ll 1} are the three main parameters in Palis-Yoccoz induction scheme.

This lemma is crucial for our purposes because it says that the exceptional set is “confined” into regions whose widths are decaying in a double exponential way to zero. Note that this is in sharp contrast with the case of the stable set of the initial horseshoe (which is confined into regions whose widths are going exponentially to zero): in other words, this lemma is a quantitative way of saying that the set {\mathcal{E}^+} is exceptional when compared with the stable lamination of the horseshoe {K_g}.

In any event, the lemma above allow us to estimate the Hausdorff {d}-measure of {\mathcal{E}^+(P_0,\dots,P_k)}. By definition, we know that a certain {g}-iterate of {\mathcal{E}^+(P_0,\dots,P_k)} is contained in the parabolic core {c(P_k)}. On the other hand, as it is shown in the figure below, we know that {c(P_k)} is contained in a vertical strip of width {\varepsilon_k:=|Q_k|^{(1-\eta)/2}|P_k|} and heigth {|Q_{k-1}|^{1/2}} (see Proposition 62 of Palis-Yoccoz article).

Fig. 2. Geometry of the parabolic core {c(P_k)} of {P_k}.

We divide this vertical strip containing {c(P_k)} into {N_k:= \frac{|Q_{k-1}|^{1/2}}{|Q_k|^{(1-\eta)/2} |P_k|}} squares of sides of lengths {\varepsilon_k} and we analyze individually their evolution under the dynamics by an inductive procedure.

Remark 1 Of course, this crude partition of {c(P_k)} into {N_k} squares of dimensions {\varepsilon_k\times\varepsilon_k} aligned along a vertical strip is motivated by the fact that we don’t want to keep track of the fine geometry of {\mathcal{E}^+(P_0,\dots,P_k)} because it gets complicated very fast (as a few iterations of the figure above shows).

More precisely, at the {i}-th step of our procedure, we have {N_{i+1}} squares of dimensions {\varepsilon_{i+1}\times\varepsilon_{i+1}} inside {Q_i}. We fix one of these squares and we note that {g^{-n_i}} sends this square into a vertical strip of width {\varepsilon_i:=\varepsilon_{i+1}|P_i|} and height {\varepsilon_{i+1}/|Q_i|:=\varepsilon_i/|P_i||Q_i|} because {(P_i,Q_i,n_i)\in\mathcal{R}} is affine-like.

Remark 2 Here, we are using the affine-like property of {g^{-n_i}} to say that essentially it is an affine map whose linear part consists of multiplying the vertical direction by {|P_i|} and the horizontal direction by {1/|Q_i|}. Evidently, this is not completely true as {g^{-n_i}} is non-linear, but the bounded distortion conditions (implicitly imposed in the definition of affine-like maps) ensure that, up to a (multiplicative) constant factor {C}, {g^{-n_i}} behaves like an affine map. Of course, this provides a reasonably good control of the geometry, but we should be careful when inductively analyzing the composition of {g^{-n_i}} for {i=1,\dots, k}. Indeed, the bounded distortion property ensures only that the composition of {g^{-n_i}} for {i=1,\dots, k} behaves like an affine map up to a multiplicative factor of {\underbrace{C \dots C}_{k}=C^k} growing exponentially with {k}. In other words, the distortion (“difference”) between {g^{-n_1-\dots-n_k}} and an affine map becomes significant as {k\rightarrow\infty}. In general, this is an important technical problem, but, fortunately, in our situation this will not be a big issue because Lemma 1 ensures that the size of the exceptional set decreases doubly exponentially fast ({\leq \varepsilon_0^{\beta^k}}) and this turns out to be sufficient to control the exponential growth of the distortion. In particular, we will ignore this minor distortion issue and we will pretend that the maps {g^{-n_i}} are affine.

Again, we divide this vertical strip into {N_i:=1/|P_i| |Q_i|} squares of sides of length {\varepsilon_i} (similarly to Figure 2). Of course, during each step of this backward inductive procedure, we need to verify the compatibility condition {\varepsilon_{i+1}<|Q_i|}. In the present case, this compatibility condition is automatically satisfied in view of the estimate of Lemma 1.

In particular, at the final step of this argument, we obtain a covering of {\mathcal{E}^+(P_0,\dots,P_k)} by a collection of {N_0=N_k/\prod\limits_{i=0}^{k-1}|P_i| |Q_i|} squares of sides of length {\varepsilon_0=\varepsilon_k\prod\limits_{i=0}^{k-1}|P_i|}. Thus, we have that

\displaystyle \inf\limits_{\substack{\mathcal{O} \textrm{ cover of } \mathcal{E}^+(P_0,\dots,P_k)  \textrm{with } \textrm{diam}(\mathcal{O})<\varepsilon_0}} \, \, \sum\limits_{O_i\in\mathcal{O}} \textrm{diam}(O_i)^{d}

\displaystyle \leq N_0\cdot\varepsilon_0^d = N_k\cdot\varepsilon_k^d\cdot\prod\limits_{i=0}^{k-1}|P_i|^{d-1}/|Q_i|

\displaystyle = |Q_k|^{\frac{(1-\eta)(d-1)}{2}}\cdot |P_k|^{d-1}\cdot \frac{|P_{k-1}|^{d-1}}{ |Q_{k-1}|^{1/2}}\cdot\prod\limits_{i=0}^{k-2}\frac{|P_i|^{d-1}}{|Q_i|}\ \ \ \ \ (1)

Since {\widetilde{\beta}(1-\eta)(d-1)>1} for any {d<2} sufficiently close to {2}, we can use Lemma 1 to see that the right-hand side of (1) can be estimated by

\displaystyle |Q_k|^{(1-\eta)(d-1)/2}\cdot |P_k|^{d-1}\cdot |Q_{k-1}|^{-1/2}\cdot |P_0|^{d-1}.

Let us take {d^->d_s^0+d_u^0-1} a real number very close to {d_s^0+d_u^0-1} and rewrite the previous expression as

\displaystyle |Q_k|^{\frac{(1-\eta)(d-1)}{2} - d^-}\cdot |Q_k|^{d^-} \cdot |P_k|^{d-1}\cdot |Q_{k-1}|^{-1/2}\cdot |P_0|^{d-1}.

Applying again Lemma 1, we can bound this expression by

\displaystyle |Q_{k-1}|^{d^*}\cdot |Q_k|^{d^-} \cdot |P_0|^{d-1}

where {d^*=\frac{3(d-1)}{2}-d^- - 1/2}. However, the hypothesis (H4) of Palis-Yoccoz article forces {0\leq d^-<1/5}, so that {d^*\geq 0} for any {d\geq 22/15}. It follows that

\displaystyle \inf\limits_{\substack{\mathcal{O} \textrm{ cover of } \mathcal{E}^+(P_0,\dots,P_k) \\ \textrm{with } \textrm{diam}(\mathcal{O})<\varepsilon_0}} \, \, \sum\limits_{O_i\in\mathcal{O}} \textrm{diam}(O_i)^{d}\leq |Q_{k-1}|^{d^*}\cdot |Q_k|^{d^-} \cdot |P_0|^{d-1}

\displaystyle \leq |Q_k|^{d^-} \cdot |P_0|^{d-1}. \ \ \ \ \ (2)

Now we use two facts derived in the pages 204 and 205 of Palis-Yoccoz paper. Firstly, they show that the number of admissible sequences {(P_0,\dots,P_k)} with fixed extremities {(P_0,Q_0,n_0)} and {(P_k,Q_k,n_k)} is {|Q_k|^{-\eta}} (see Equation (11.77) of their article). Secondly, the sum {\sum_{Q_k} |Q_k|^{d^- - \eta}} is bounded because the results of the Subsection 11.5.9 of their article show that {\sum\limits_{Q \textrm{ critical }} |Q|^{d^- - \eta}} converges provided that {d^--\eta>d_s+d_u-1}.

Remark 3 It is hidden in the estimate{\sum\limits_{Q \textrm{ critical }} |Q|^{d^- - \eta}<\infty} for {d^--\eta>d_s+d_u-1} the fundamental (moral) fact that the critical locus is expected to have Hausdorff dimension {d_s+d_u-1} that we alluded to in the introduction of the previous post of this series.

Putting these two facts together with (2), we see that the “Hausdorff {d}-measure of {\mathcal{E}^+=\bigcup\limits_{(P_0,\dots,P_k) \textrm{ admissible}}\mathcal{E}^+(P_0,\dots,P_k)} at scale {\varepsilon_0=\varepsilon_0(k)}” satisfies

\displaystyle \inf\limits_{\substack{\mathcal{O} \textrm{ cover of } \mathcal{E}^+ \\ \textrm{with } \textrm{diam}(\mathcal{O})<\varepsilon_0}} \, \, \sum\limits_{O_i\in\mathcal{O}} \textrm{diam}(O_i)^{d}\leq \sum\limits_{P_0,\dots,P_k} |Q_k|^{d^-}\leq \sum\limits_{Q_k} |Q_k|^{d^- -C\eta}\leq C.

Because {\varepsilon_0=\varepsilon_0(k)\rightarrow0} as {k\rightarrow\infty}, this proves that {\textrm{HD}(\mathcal{E}^+)\leq d<2}. Hence, the proof of Theorem 1 is complete.

2. Final comments on further results

The attentive reader observed that the arguments of the previous subsection were based on soft analysis of the geometry of the exceptional set. Indeed, every time the shape of the parabolic cores {c(P_i)} was ready to get complicated, we divide it into squares and we analyzed the evolution of individual squares. In particular, every time we saw some parabolic geometry, we covered the “tip of the parabola” by a black box (square) and we forgot about the finer details of {\mathcal{E}^+} in this region. Of course, it is not entirely surprising that this kind of soft estimate works to show {\textrm{HD}(W^s(\Lambda_g))<2}, but it is too crude if one wishes to compute the actual value of {\textrm{HD}(W^s(\Lambda_g))}.

In particular, if one desires to prove that {\mathcal{E}^+} is really exceptional so that {\textrm{HD}(W^s(\Lambda_g))} is close to the expected dimension {1+d_s^0}, one has to somehow “face” the geometry of {\mathcal{E}^+} and its successive passages through parabolic cores {c(P_i)}.

In the forthcoming article by Jacob, Jean-Christophe and myself, we improve the soft strategy above without entering too much into the fine geometry of {\mathcal{E}^+} by noticing that each {\mathcal{E}^+(P_0,\dots,P_k)} is the image of the parabolic core {c(P_k)} with a map of the form {g^{-n_0}\circ G\circ g^{-n_1}\circ\dots} (obtained by alternating compositions of affine-like iterates of {g} and the folding map {G}) whose derivative and Jacobian can be reasonably controlled. Using this control, we can study the Hausdorff {d}-measure of {\mathcal{E}^+} at certain fixed scales {\varepsilon_0=\varepsilon_0(k)} in terms of the geometry of {c(P_k)} and the bounds on the derivative and the Jacobian of the map sending {c(P_k)} into {\mathcal{E}^+}. By putting forward this estimate, we can show that {W^s(\Lambda_g)} has the expected Hausdorff dimension (namely {1+d_s}) in a certain subregion {\mathcal{D}} of values of stable and unstable dimensions {d_s^0} and {d_u^0} of the initial horseshoe. In Figure 3 below we depicted in wave texture the region {\mathcal{D}\cap\{(d_s^0,d_u^0): d_s^0\leq d_u^0\}} inside the larger region {\{(d_s^0,d_u^0): \textrm{Palis-Yoccoz condition } (3) \textrm{ below holds}\}} where

\displaystyle (d_s^0+d_u^0)^2+\max(d_s^0,d_u^0)^2< (d_s^0+d_u^0)+\max(d_s^0,d_u^0) \ \ \ \ \ (3)

Actually, in this picture we drew only {\mathcal{D}\cap\{(d_s^0,d_u^0): d_s^0\leq d_u^0\}} because the other half {\mathcal{D}\cap\{(d_s^0,d_u^0): d_s^0\geq d_u^0\}} of {\mathcal{D}} can be deduce by symmetry.

Fig. 3. {\mathcal{D}\cap\{d_s^0\leq d_u^0\}} in wave texture sitting inside {\{(d_s^0,d_u^0): (3) \textrm{ holds}\}}.

The intersection {\mathcal{D}\cap\{(d_s^0,d_u^0):d_s^0=d_u^0\}} of the region {\mathcal{D}} with the diagonal (corresponding to the “conservative case”) can be explicitly computed:

\displaystyle \mathcal{D}\cap\{(d_s^0,d_u^0):d_s^0=d_u^0\}=\{(x,x): 1/2<x<0.545...\}

Remark 4 The nomenclature “conservative” comes from the fact that the stable and unstable dimensions of any horseshoe of a area-preserving diffeomorphism coincide (see, e.g., this article of H. McCluskey and A. Manning).

In particular, by putting this together with Figure 3, we see that {\mathcal{D}} occupies slightly less than half of region given by Palis-Yoccoz condition (3).

Finally, let us remark that we get the expected Hausdorff dimensions for {W^s(\Lambda_g)} and {W^u(\Lambda_g)} (in region {\mathcal{D}}), but the arguments can’t be used to get the expected Hausdorff dimension for {\Lambda_g=W^s(\Lambda_g)\cap W^u(\Lambda_g)}. In fact, our constructions so far start from the future of {W^s(\Lambda_g)} and the past of {W^u(\Lambda_g)} where some geometric control is available, e.g., in the form of nice partitions, and then it tries to bring back the information, i.e., partitions, by analyzing the {g}-iterates used in our way back to the present time. Of course, this works if we deal separately with the past or the future, but if we try to deal with both at the same time, we run into trouble because it is not obvious how the partitions coming from the future and the past intersect in the present time (due to the lack of transversality produced by {g}-iterates related to the folding map {G}). Evidently, the question of getting the expected Hausdorff dimension for {\Lambda_g} is natural and interesting, and we hope to address this issue in our forthcoming paper.


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