Posted by: matheuscmss | October 28, 2011

Typical smooth cocycles over a hyperbolic basis have non-zero Lyapunov exponents II

Today’s post brings the lectures notes I used in my second (and last) exposition on Marcelo Viana’s article “Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents” (as a part of a “Groupe de Travail” organized by S. Crovisier and J. Buzzi).

In this previous post, we started with a ${C^{r,\nu}}$ cocycle ${A:M\rightarrow SL(d,\mathbb{K})}$ (${r\in\mathbb{N}}$, ${\nu=1}$) over a non-uniformly hyperbolic basis ${(f,\mu)}$. Then, we saw that the vanishing of the top exponent ${\lambda^+(A,x)=0}$ at ${\mu}$-a.e. ${x}$ (that is the same as all Lyapunov exponents vanish because ${A}$ takes values in ${SL(d,\mathbb{K})}$) implies some sort of domination (i.e., the dynamics of the cocycle along the fibers is dominated by the dynamics of the basis) and this last property (actually, 3-domination) allows for the construction of stable and unstable holonomies, e.g., ${H^{s}_{A,y,z}:=\lim\limits_{n\rightarrow+\infty} A^{-n}(z)A^n(y)}$, ${y,z\in W^s_{loc}(x)}$, depending in a Lispchitz way on ${y,z}$ and in a ${C^1}$ way on ${A}$. Before putting these holonomies to work, we will state (without proof) the following result justifying the terminology “holonomy” (but which will be not used in the sequel):

Lemma 1 Let ${x}$ be a ${1}$-dominated point, say ${x\in\mathcal{H}(K,\tau)\cap D_A(N,\theta)}$ with ${\theta<\tau}$. For any ${y\in W^s_{loc}(x)}$ and ${\xi\in\mathbb{P}(\mathbb{K}^d)}$, one has

• ${\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log\textrm{dist}(f_A^n(x,\xi),f_A^n(y,h_{x,y}^s(\xi)))\leq-\tau}$;
• ${\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\log\textrm{dist}(f_A^n(x,\xi),f_A^n(y,\eta))<-\theta}$ if and only if ${\eta=h_{x,y}^s(\xi)}$.

Here, as usual, ${f_A:M\times\mathbb{P}(\mathbb{K}^d)\rightarrow M\times\mathbb{P}(\mathbb{K}^d)}$ is the projective cocycle associated to ${F_A:M\times\mathbb{K}^d\rightarrow M\times \mathbb{K}^d}$, ${F_A(x,v)=(f(x),A(x)v)}$.

For later use, we denote by ${h^{s/u}_{y,z} = h^{s/u}_{A,y,z}}$ the projective stable/unstable holonomy associated to the stable/unstable holonomy ${H^{s/u}_{y,z} = H^{s/u}_{A,y,z}}$.

Now, we proceed to discuss the 2nd step of the strategy of proof of Theorem 5 of the first post on Marcelo’s article whose statement we recall below:

Theorem 5. For all ${r\in\mathbb{N}}$ and ${0\leq\nu\leq 1}$ with ${r+\nu>0}$, and ${\mu}$ ergodic measure with local product structure, the set ${\mathcal{A}}$ of cocycles ${F\in\mathcal{S}^{r,\nu}(f,\mathcal{E})}$ with ${\lambda^+(F,x)>0}$ for ${\mu}$-a.e. ${x}$ is open, dense and it has ${\infty}$-codimension.

2nd step of the proof of Theorem 5: holonomies and projective cocycles

We start our considerations with the following notion:

Definition 2 We say that a compact set ${\mathcal{O}\subset\mathcal{H}(K,\tau)\cap D_A(N,\theta)}$, ${3\theta<\tau}$, is a holonomy block.

During this entire section, we’ll make the following assumption:

$\displaystyle \textrm{(ZE)} \quad \lambda^+(A,x)=0 \quad \textrm{for } \mu-\textrm{a.e. } x$

Now, we take ${m}$ a ${f_A}$-invariant probability measure on ${M\times\mathbb{P}(\mathbb{K}^d)}$ projecting to ${\mu}$ (such a ${\mu}$ always exists by compactness of ${\mathbb{P}(\mathbb{K}^d)}$). By Rokhlin’s disintegration theorem (see also this link here for a modern exposition of this theorem), ${m}$ can be disintegrated into a family ${\{m_z: z\in M\}}$ of probabilities on ${\mathbb{P}(\mathbb{K}^d)}$ such that

$\displaystyle m(E) = \int m_z(E\cap (\{z\}\times\mathbb{P}(\mathbb{K}^d))) d\mu(z).$

Moreover, Rokhlin’s theorem also ensures that the disintegration is essentially uniquein the sense that if ${\{\widetilde{m}_z: z\in M\}}$ also verifies

$\displaystyle m(E) = \int \widetilde{m}_z(E\cap (\{z\}\times\mathbb{P}(\mathbb{K}^d))) d\mu(z),$

then ${\widetilde{m}_z = m_z}$ for ${\mu}$-a.e. ${z\in M}$.

Next, let us fix a holonomy block ${\mathcal{O}}$ of positive ${\mu}$-measure, say ${\mathcal{O}\subset\mathcal{H}(K,\tau)}$, and let ${\delta=\delta(K,\tau)>0}$ be a small positive real number such that ${\#(W^s_{loc}(x)\cap W^u_{loc}(y))=1}$ for every ${x,y\in\mathcal{H}(K,\tau)}$ with ${\textrm{dist}(x,y)\leq\delta}$. Here, we denote by ${[x,y]}$ the unique point ${[x,y]\in W^s_{loc}(x)\cap W^u_{loc}(y)}$ whenever ${\textrm{dist}(x,y)\leq\delta}$, ${x,y\in\mathcal{H}(K,\tau)}$. For each ${\overline{x}\in\textrm{supp}(\mu)}$, we define

$\displaystyle \mathcal{N}_{\overline{x}}^s(\delta):=\mathcal{N}_{\overline{x}}^s(K,\tau,\delta) = \{z\in W^s_{loc}(\overline{x})\cap W^u_{loc}(y): \textrm{dist}(y,\overline{x})\leq\delta, y\in\mathcal{H}(K,\tau)\},$

$\displaystyle \mathcal{N}_{\overline{x}}^u(\delta):=\mathcal{N}_{\overline{x}}^u(K,\tau,\delta) = \{z\in W^u_{loc}(\overline{x})\cap W^s_{loc}(y): \textrm{dist}(y,\overline{x})\leq\delta, y\in\mathcal{H}(K,\tau)\}$

and

$\displaystyle \mathcal{N}_{\overline{x}}(\delta)=[\mathcal{N}_{\overline{x}}^s(\delta),\mathcal{N}_{\overline{x}}^u(\delta)].$

Similarly, we define ${\mathcal{N}_{\overline{x}}^s(\mathcal{O},\delta)}$, ${\mathcal{N}_{\overline{x}}^s(\mathcal{O}, \delta)}$ and ${\mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)}$ by replacing the condition ${y\in\mathcal{H}(K,\tau)}$ by ${y\in\mathcal{O}}$ in the definitions above.

The main result of this section is the following proposition:

Proposition 3 (Invariance by holonomies) Under the assumption (ZE), there exists a set ${E^s\subset\mathcal{N}_{\overline{x}}(\mathcal{O},\delta)}$ of full ${\mu}$-measure such that

$\displaystyle m_{z_2} = (h^s_{z_1,z_2})_* m_{z_1}$

for all ${z_1,z_2\in E^s}$ in a fixed stable leaf ${[z,\mathcal{N}_{\overline{x}}^s(\delta)]}$.

As we hinted in the description of the strategy of proof of Theorem 5 (in this previous post here), the key tool in the proof of this proposition is the following result of F. Ledrappier (whose proof will be omitted):

Theorem 4 (F. Ledrappier) Let ${T:M_*\rightarrow M_*}$ be an injective measurable transformation of a Lebesgue space${(M_*,\mathcal{M}_*,\mu_*)}$, and ${B:M_*\rightarrow GL(d,\mathbb{C})}$ a ${\log}$-integrable cocycle (i.e., ${\log\|B^{\pm1}\|}$ is ${\mu}$-integrable). Let ${\mathcal{B}\subset\mathcal{M}_*}$ is a ${\sigma}$-algebra such that

• ${T^{-1}(\mathcal{B})\subset\mathcal{B}}$ (mod ${0}$) and ${\{T^n(\mathcal{B}): n\in\mathbb{Z}\}}$ generates ${\mathcal{M}_*}$ (mod ${0}$);
• the ${\sigma}$-algebra generated by ${B}$ is contained in ${\mathcal{B}}$ (mod ${0}$).

Then, if ${\lambda^-(B,x) = \lambda^+(B,x)}$ for ${\mu_*}$-a.e. ${x}$, we have that the map ${z\mapsto m_z}$ is ${\mathcal{B}}$-measurable (mod ${0}$), where ${\{m_z: z\in M_*\}}$ is the disintegration of a ${f_B}$-invariant probability ${m}$ on ${M_*\times\mathbb{P}(\mathbb{C}^d)}$ (and ${f_B}$ is the projective cocycle corresponding to ${T}$ and ${B}$).

Once one disposes of Ledrappier’s theorem, the idea of the proof of Proposition 3 is quite simple: one “translates” the invariance by holonomies condition into a measurability statement with respect to an adequate ${\sigma}$-algebra. To formalize this idea, we need the following “Markov partition like statement”:

Proposition 5 There exists an integer ${N\geq 1}$ and a family of subsets ${\{S(z): z\in\mathcal{N}_{\overline{x}}(\delta)\}}$ such that

• The subsets ${S(z)}$ are not very small: ${[z,\mathcal{N}_{\overline{x}}^s(\delta)]\subset S(z)\subset W^s_{loc}(z)}$ for all ${z\in\mathcal{N}_{\overline{x}}^u(\delta)}$;
• Markov-like property: for any ${\ell\geq 1}$, ${z,\zeta\in\mathcal{N}_{\overline{x}}^u(\delta)}$, if ${f^{\ell N}(S(\zeta))\cap S(z)\neq \emptyset}$, then ${f^{\ell N}(S(\zeta))\subset S(z)}$.

At this point, I asked the audience of my second talk whether they wanted to hear about the details of this proposition. After a little discussion, Sylvain and Francois voted for skipping it since the arguments are very close to Bowen’s classical construction of Markov partitions (for uniformly hyperbolic systems), and I ended up by accepting their (sensate) decision. However, for this blog post, I believe that we could take the opportunity to discuss this topic, and we’ll do so here. But, in order to avoid interrupting the proof of Theorem 5, we’ll postpone it to the “Appendix” (last section of this post).

Anyway, by assuming this proposition, we consider ${\{S(z): z\in\mathcal{N}_{\overline{x}}(\mathcal{O},\delta)\}}$ as above, and, for each ${z\in\mathcal{N}_{\overline{x}}^u(\mathcal{O},\delta)}$, denote by ${r(z)\in\mathbb{N}\cup\{\infty\}}$ the biggest ${j}$ such that ${f^j(S(z))\cap \bigcup\limits_{w\in\mathcal{N}_{\overline{x}}^u(\mathcal{O},\delta)} S(w)=\emptyset}$. Observe that, without loss of generality, we can also assume that ${N=1}$ in the proposition above.

Let ${\mathcal{B}}$ be the ${\sigma}$-algebra generated by ${\{f^j(S(z)): z\in \mathcal{N}_{\overline{x}}(\mathcal{O},\delta), 0\leq j\leq r(z)\}}$, i.e.,

$\displaystyle \mathcal{B}=\{E\subset M \textrm{ measurable}: \forall\, z,j, E\supset f^{j}(S(z)) \textrm{ or } E\cap f^j(S(z))=\emptyset\}$

Finally, let us introduce the cocycle ${B:M\rightarrow GL(d,\mathbb{C})}$,

$B(x) = \left\{\begin{array}{cl} A(f^j(z)) = H^s_{f(x),f^{j+1}(z)}A(x)H^s_{f^j(z),x}, & \textrm{if } x\in f^j(S(z)), z\in\mathcal{N}_{\overline{x}}^u(\mathcal{O},\delta), 0\leq j \\ H^s_{f(x),w}A(x)H^s_{f^{r(z)}(z),x}, & \textrm{if } x\in f^{r(z)}(S(z)) \textrm{ and } f^{r(z)+1}(S(z))\subset S(w), \\ A(x) & \textrm{otherwise}.\end{array}\right.$

The lemma allowing us to convert the invariance by holonomies statement for ${A}$ into a measurability statement for ${B}$ is the following one:

Lemma 6 We have the following properties:

• (i) ${f^{-1}(\mathcal{B})\subset\mathcal{B}}$ and ${\{f^n(\mathcal{B}): n\in\mathbb{N}\}}$ generates the Borel ${\sigma}$-algebra ${\mathcal{M}_*}$ of ${M}$;
• (ii) the ${\sigma}$-algebra generated by ${B}$ is contained in ${\mathcal{B}}$;
• (iii) ${\log\|B^{\pm1}\|}$ is ${\mu}$-integrable;
• (iv) the cocycles ${A}$ and ${B}$ have the same Lyapunov exponents.

Proof: Note that ${f(\mathcal{B})}$ is the ${\sigma}$-algebra generated by ${\{f^{j+1}(S(z)): z\in\mathcal{N}_{\overline{x}}(\mathcal{O},\delta), 0\leq j\leq r(z)\}}$. By the Markov-like property (see the 2nd item of Proposition 5), ${f(\mathcal{B})\supset\mathcal{B}}$, i.e., ${f^{-1}(\mathcal{B})\subset \mathcal{B}}$. Also, since ${f^n(\mathcal{B})}$ is the ${\sigma}$-algebra generated by ${\{f^{n+j}(S(z)): z\in\mathcal{N}_{\overline{x}}(\mathcal{O},\delta), 0\leq j\leq r(z)\}}$, and ${\textrm{diam}(f^{n+j}(S(z)))\leq Ce^{-\tau n}\rightarrow 0}$ as ${n\rightarrow\infty}$ (as ${S(z)\subset W^s_{loc}(z)}$, ${z\in\mathcal{H}(K,\tau)}$). It follows that ${\{f^n(\mathcal{B}): n\in\mathbb{N}\}}$ generates ${\mathcal{M}_*}$. This proves (i) above.

By definition of ${B}$, one can check that ${B^{-1}(E)\in\mathcal{B}}$ whenever ${E\subset GL(d,\mathbb{C})}$ is measurable (this is essentially the fact that ${B}$ is constant equal to ${A(f^j(z))}$ on each ${f^j(S(z))}$ with ${0\leq j). In other words, the ${\sigma}$-algebra generated by ${B}$ is contained in ${\mathcal{B}}$. This proves (ii) above.

The ${\mu}$-integrability of ${\log\|B^{\pm1}\|}$ is “almost obvious” from the ${\mu}$-integrability of ${\log\|A^{\pm1}\|}$, except maybe by the case ${x\in f^{r(z)}(S(z))}$ and ${f^{r(z)+1}(S(z))\subset S(w)}$. In the latter case, one notices that ${H^s_{f^{r(z)}(z),f^{r(z)}(\zeta)}}$ and ${H^s_{f^{r(z)+1}(\zeta),w}}$ are uniformly close to the identity (by Proposition 12 and Corollary 14 of the previous post). This proves (iii) above.

Finally, it suffices to show that the cocycles ${A}$ and ${B}$ are conjugated by a conjugacy at a bounded distance of the identity to get that they have the same Lyapunov exponents. In our case, we have ${B(x) = H(f(x))\circ A(x)\circ H(x)^{-1}}$ where

$H(y) = \left\{\begin{array}{cl}H_{y,f^j(z)}^s, & \textrm{if }y\in f^j(S(z)), 0\leq j \\ \textrm{Id}, & \textrm{otherwise}.\end{array}\right.$

Again by Proposition 12 and Corollary 14 of the previous post, we have that the conjugation ${H(y)}$ is uniformly close to the identity. This proves (iv). $\Box$

From this lemma, we can translate the invariance by holonomy condition for ${A}$ into a measurability condition for ${B}$ to get a proof of Proposition 3 as follows:

Proof: In the case of a ${SL(d,\mathbb{K})}$-valued cocycle (${\mathbb{K}=\mathbb{R}, \mathbb{C}}$), one has that ${\lambda^+(A,x)=0}$ for ${\mu}$-a.e. ${x}$ if and only if ${\lambda^+(A,x)=\lambda^-(A,x)}$ for ${\mu}$-a.e. ${x}$. By the previous lemma, this implies that ${\lambda^+(B,x)=\lambda^-(B,x)}$ for ${\mu}$-a.e. ${x}$. Also, we observe that, by Rokhlin’s theorem, ${m}$ is ${f_A}$-invariant if and only ${m_{f(x)}=A(x)_* m_x}$ for ${\mu}$-a.e. ${x}$.

Now, we construct the following family of probabilities ${\{\widetilde{m}_x: x\in M\}}$:

$\widetilde{m}_x = \left\{\begin{array}{rl} (h^s_{x,f^j(z)})_* m_x, & \textrm{if } x\in f^j(S(z)), z\in\mathcal{N}_{\overline{x}}(\mathcal{O},\delta), 0\leq j\leq r(z) \\ m_x, & \textrm{otherwise}\end{array}\right.$

We claim that the probability measure ${\widetilde{m}}$ on ${M\times\mathbb{P}(\mathbb{K}^d)}$ with disitegration ${\{\widetilde{m}_x: x\in M\}}$ is ${f_B}$-invariant. Indeed,

• if ${x\in f^j(S(z))}$, ${0\leq j , we have

$\displaystyle B(x)_* \widetilde{m}_x:=(h^s_{f(x),f^{j+1}(z)})_* A(x)_* m_x = (h^s_{f(x),f^{j+1}(z)})_* m_{f(x)} = \widetilde{m}_{f(x)}.$

• if ${x\in f^{r(z)}(S(z))}$, ${f^{r(z)+1}(S(z))\subset S(w)}$,

$\displaystyle B(x)_*\widetilde{m}_x = (h_{f(x),w}^s)_* A(x)_* m_x = (h_{f(x),w}^s)_* m_{f(x)} = \widetilde{m}_{f(x)}.$

• the remaining case is evident (as ${B(x)=A(x)}$ and ${\widetilde{m}_x = m_x}$).

In any case, we see that ${\widetilde{m}}$ is ${f_B}$-invariant. This discussion combined with Lemma 6 say that the cocycle ${B}$ and the ${\sigma}$-algebra ${\mathcal{B}}$ fulfill the hypothesis of Ledrappier’s theorem. Hence, we conclude that the map

$\displaystyle x\mapsto\widetilde{m}_x$

is ${\mathcal{B}}$-measurable. By definition of ${\mathcal{B}}$, this ${\mathcal{B}}$-measurability statement implies the following property: there is a subset ${E^s\subset\mathcal{N}_{\overline{x}}(\mathcal{O},\delta)}$ of full ${\mu}$-measure such that

$\displaystyle \widetilde{m}_{z_1} = \widetilde{m}_{z_2}$

whenever ${z_1,z_2\in E^s\cap S(z)}$. By definition of ${\widetilde{m}_x}$ and the nice “composition” properties of the stable holonomies, we have that the preivous equality means

$\displaystyle (h^s_{z_1,z})_*m_{z_1} = (h^s_{z_2,z})_*m_{z_2} \quad \textrm{i.e.} \quad (h^s_{z_1,z_2})_*m_{z_1} = m_{z_2}$

whenever ${z_1,z_2\in E^s\cap S(z)}$. Since ${S(z)\supset [z,\mathcal{N}_{\overline{x}}^s(\delta)]}$ (i.e., ${S(z)}$ is not very small, see Proposition 5), the proof of Proposition 3 is complete. $\Box$

Next, we combine Proposition 3 of invariance by holonomy with the local product structure of ${\mu}$ to deduce the following continuity result:

Proposition 7 Under the assumption (ZE), it holds that any ${f_A}$-invariant probability ${m}$ (projecting to ${\mu}$) admits a disintegration ${\{\widetilde{m}_z: z\in M\}}$ such that

• (a) the map ${\textrm{supp}(\mu|_{\mathcal{N}_{\overline{x}}(\mathcal{O},\delta)})\ni z\mapsto \widetilde{m}_z}$ is continuous;
• (b) ${\widetilde{m}_z}$ is invariant under stable and unstable holonomies on the whole support of ${\mu|_{\mathcal{N}_{\overline{x}}(\mathcal{O},\delta)}}$.

Proof: Let ${E=E^s\cap E^u}$, where ${E^s}$ and ${E^u}$ are the subsets provided by Proposition 3. We have that ${\mu(\mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)-E)=0}$. Since ${\mu}$ has local product structure (${\mu\sim\mu^s\times\mu^u}$), we have that ${\mu^s([\xi,\mathcal{N}_{\overline{x}}^s(\mathcal{O}, \delta)]\cap (\mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)-E)) = 0}$ for ${\mu^u}$-a.e. ${\xi\in\mathcal{N}_{\overline{x}}^u(\mathcal{O}, \delta)}$. Fix ${\xi}$ a ${\mu^u}$-generic point and define

$\displaystyle \overline{m}_z = (h^u_{\eta,z})_* m_{\eta}$

when ${z\in [\mathcal{N}_{\overline{x}}^u(\mathcal{O}, \delta),\eta]}$ and ${\eta\in [\xi,\mathcal{N}_{\overline{x}}^s(\mathcal{O}, \delta)]}$, and ${\overline{m}_z=m_z}$ otherwise. (This task would be easier if we could take ${\xi=\overline{x}}$, but we can’t do that in general). By definition of ${E^u}$ and local product structure, ${\overline{m}_z=m_z}$ for ${\mu}$-a.e. ${z}$, i.e., ${\overline{m}_z}$ is a disintegration of ${m}$. Moreover, ${\overline{m}_z}$ varies continuously along unstable leaves ${[\mathcal{N}_{\overline{x}}^s(\mathcal{O}, \delta), \eta]}$ because, for each ${\eta}$, the unstable holonomy ${h^u_{\eta,z}}$ is a Lipschitz function of ${z}$ (see Proposition 12 of the previous post). Now, we fix ${\eta}$ a ${\mu^s}$-generic point and let ${\{m_z^s: z\in M\}}$ the disintegration of ${m}$ obtained from ${\overline{m}_z}$ by forcing stable-holonomy invariance from ${[\mathcal{N}_{\overline{x}}^s(\mathcal{O}, \delta),\eta]}$, i.e.,

$\displaystyle m_z^s = (h^s_{\zeta,z})_* \overline{m}_z$

when ${z\in [\zeta,\mathcal{N}_{\overline{x}}^u(\mathcal{O}, \delta)]}$ and ${\zeta\in [\mathcal{N}_{\overline{x}}^s(\mathcal{O}, \delta),\eta]}$, and ${m_z^s=\overline{m}_z}$ otherwise. Again by definition of ${E^s}$ and local product structure, it follows that ${\{m_z^s: z\in M\}}$ is a disintegration of ${m}$ such that ${m_z^s}$ is invariant by stable holonomies and ${m_z^s}$ varies continuously on the whole${\mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)}$!

By a dual procedure, we can also construct a disintegration ${\{m_z^u: z\in M\}}$ of ${m}$ such that ${m_z^u}$ is invariant by unstable holonomies and ${m_z^u}$ varies constinuously on the whole ${\mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)}$. By the uniqueness of the disintegration, we have that ${m_z^s=m_z^u}$ for ${\mu}$-a.e. ${z\in \mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)}$. By continuity, actually we must have ${m_z^s=m_z^u}$ for every ${z\in\mathcal{N}_{\overline{x}}(\mathcal{O}, \delta)}$. This completes the proof. $\Box$

3rd step of the proof of Theorem 5: contamination by periodic points (part I)

Recall (from the previous post) that a periodic point ${p}$ of ${f}$ is dominated for ${A}$ when ${p\in\mathcal{H}(K,\tau)\cap D_A(N,\theta)}$ with ${3\theta<\tau}$.

Proposition 8 If ${p}$ is a periodic point dominated for ${A}$, then there exists a neighborhood ${\mathcal{U}}$ of ${A}$ such that, for any ${z\in W^s_{loc}(p)}$, the map

$\displaystyle \mathcal{U}\ni B\mapsto h^s_{B,p,z}$

is ${C^1}$. Moreover, given ${\xi_1,\dots,\xi_d\in\mathbb{P}(\mathbb{K}^d)}$ linearly independent, the map

$\displaystyle \mathcal{U}\ni B\mapsto (h^s_{B,p,z}(\xi_1),\dots, h^s_{B,p,z}(\xi_d))\in \mathbb{P}(\mathbb{K}^d)^d$

is a submersion.

For the proof of this proposition, we will need the following fact about the Lie group ${SL(d,\mathbb{K})}$, ${\mathbb{K}=\mathbb{R},\mathbb{C}}$:

Remark 1 Given ${\eta_1,\dots,\eta_d\in \mathbb{P}(\mathbb{K}^d)}$ linearly independent, the map ${SL(d,\mathbb{K})\rightarrow \mathbb{P}(\mathbb{K}^d)^d}$,

$\displaystyle \beta\mapsto (\beta(\eta_1),\dots,\beta(\eta_d))$

is a submersion.

Using this remark, the proof of Proposition 8 goes as follows:

Proof: Fix ${U}$ be a neighborhood of ${z}$ such that ${U\cap\{f^j(p), f^j(z): j\geq 1\}=\emptyset}$. By the usual partition of unity argument, for any ${\beta\in T_{B(z)}SL(d,\mathbb{K})}$ (i.e., a traceless ${d\times d}$ matrix), there exists ${\dot{B}\in T_B\mathcal{S}^{r,\nu}}$ such that ${\dot{B}(z)=\beta}$ and ${\dot{B}(w)=0}$ for all ${w\notin U}$. By the formula of Lemma 15 of the previous post (see also the remark below), the choice of ${U}$ and the fact that ${\dot{B}(w)=0}$ when ${w\notin U}$ imply

$\displaystyle \partial_B H^s_{B,p,z}\dot{B} = -B(z)^{-1}\dot{B}(z) H^s_{B,p,z}$

Since the directions ${\eta_i=H^s_{B,p,z}(\xi_i)}$ are linearly independent (because ${\xi_i}$ are linearly independent), the proposition follows from Remark 1. $\Box$

Remark 2 In the previous argument, the computation of derivative of ${H^s_{B,p,z}}$ with respect to the variable ${B}$ (provided by the formula of Proposition bla of the previous post) was simplified by the choice of ${\dot{B}}$ with ${\dot{B}(w)=0}$ for all ${w\notin U}$. However, this choice of ${\dot{B}}$ depends on a partition of unity argument, and hence it is not well suited to deal with real/complex-analyticcocycles (even though it is well-adapted to ${C^{r,\nu}}$ cocycles). In principle, one could work with “more global” choices of ${\dot{B}}$, but this amounts to deal with the complete expression of ${\partial_B H^s_{B,p,z}\dot{B}}$, namely,

$\displaystyle \partial_B H_{B,y,z}^s(\dot{B}) = \sum\limits_{i=0}^{\infty} B^i(z)^{-1} [H_{B,f^i(y),f^i(z)}^s B(f^i(y))^{-1} \dot{B}(f^i(y)) - B(f^i(z))^{-1}\dot{B}(f^i(z))H_{B,f^i(y),f^i(z)}^s] B^i(y)$

and it is not clear how to use this to get that the desired map is a submersion. Actually, to the best of my knowledge, the question of extending Marcelo’s theorem to real/complex-analytic cocycles is open, and, as far as I can see, the choice of “good” ${\dot{B}}$‘s (with respect to the formula for ${\partial_B H^s_{B,p,z}\dot{B}}$) is the sole issue here.

A corollary of the proof of Proposition 8 is the following result:

Corollary 9 Given ${p_1,\dots,p_k}$ periodic points of ${f}$ with periods ${\kappa_1,\dots,\kappa_k}$ (resp.), the map

$\displaystyle \mathcal{S}^{r,\nu}\ni A\mapsto (A^{\kappa_1}(p_1),\dots,A^{\kappa_k}(p_k))\in SL(d,\mathbb{K})^k$

is a submersion.

Proof: For each ${j=1,\dots,k}$, let ${\beta_j:=A^{\kappa_j}(p_j)}$. Given ${\dot{\beta}_j}$ in ${T_{\beta_j} SL(d,\mathbb{K})}$, we fix ${U_j}$ a neighborhood of ${p_j}$ such that they are mutually disjoint and ${p_j}$ is the unique point in the intersection of ${U_j}$ and the several periodic orbits. By the usual partition of unity argument, we can take a ${C^1}$ curve ${(-\varepsilon,\varepsilon)\ni t\mapsto A(t)\in \mathcal{S}^{r,\nu}}$ such that

• ${A(0)=A}$ and ${A(t)(w)=A(w)}$ for every ${w\notin U_1\cup\dots\cup U_k}$, ${t\in(-\varepsilon,\varepsilon)}$;
• ${\partial_t A(t)(p_j)|_{t=0}=A^{-\kappa_j+1}(p_j)\dot{\beta}_j}$ for each ${j=1,\dots,k}$.

It follows that ${\partial_t A(t)^{\kappa_j}(p_j)|_{t=0} = A^{\kappa_j-1}(f(p_j))\partial_t A(t)(p_j)|_{t=0} = \dot{\beta}_j}$. Since ${\dot{\beta}_j}$ are arbitrary, the proof of the corollary is complete. $\Box$

The discussion above prepares the “contamination by periodic points” argument: roughly speaking, we just saw that one has sufficient freedom to deform ${C^{r,\nu}}$ cocycles nearby any finite family of periodic points. Of course, in order to profit of this “freedom”, we need to produce “lots” of periodic points. This will be the main concern of the remainder of this section (where we will construct holonomy blocks with “several” periodic points).

In the sequel, we denote by ${M_0=\{x\in M: \lambda^+(A,x)=0\}}$ and we assume that ${\mu(M_0)>0}$.

Proposition 10 For every ${\varepsilon>0}$, ${\ell\geq 1}$, there exists a holonomy block ${\widetilde{\mathcal{O}}}$ of ${A}$ such that ${\mu(M_0-\widetilde{\mathcal{O}})<\varepsilon}$ and there are ${p_1,\dots,p_{\ell}\in\widetilde{\mathcal{O}}}$ distinct dominated periodic points with

• (1) the periodic points are heteroclinically related: ${\#(W^u_{loc}(p_i)\cap W^s_{loc}(p_j))=1}$
• (2) the periodic points are “generic”: ${p_i\in\textrm{supp}(\mu|_{\widetilde{\mathcal{O}}\cap f^{-\kappa_i}(\widetilde{\mathcal{O}})})}$, where ${\kappa_i}$ is the period of ${p_i}$.

The proof of this proposition is an adaptation of the so-called Katok’s shadowing lemma allowing to construct “lots” of periodic points inside hyperbolic blocks for non-uniformly hyperbolic systems:

Theorem 11 (Katok’s shadowing lemma) For every ${j\geq 1}$ and ${\gamma>0}$, there are ${K>0}$, ${\tau>0}$, ${\rho>0}$ and compact subsets ${\mathcal{K}_j}$ with ${\mu(K_j)\rightarrow 1}$ as ${j\rightarrow\infty}$ such that for any ${z\in\mathcal{K}_j}$ and ${\kappa\geq 1}$ with ${f^{\kappa}(z)\in\mathcal{K}_j}$ and ${\textrm{dist}(f^{\kappa}(z),z)<\varepsilon}$, there is a periodic point ${p\in M}$ of period ${\kappa}$ verifying the following properties:

• ${p}$ is hyperbolic, ${|\log|\alpha_s||>\kappa\tau}$ for each ${\alpha_s}$ eigenvalue of ${Df^{\kappa}(p)}$, and ${\textrm{dist}(f^n(x),f^n(y))\leq K e^{-\tau n} \textrm{dist}(x,y)}$ for all ${n\geq 0}$, ${x,y\in W^s_{loc}(p)}$;
• ${W^s_{loc}(p)}$ has size ${>\rho}$ and it is uniformly transverse (i.e., it makes an angle ${>\rho}$) with all unstable disks centered at ${w\in\mathcal{K}_j}$; in particular, ${\#(W^s_{loc}(p)\cap W^u_{loc}(w))=1}$ for all ${w\in\mathcal{K}_j}$;
• ${\textrm{dist}(f^j(p),f^j(z))<\gamma}$ for each ${j=0,\dots,\kappa}$.

Morally speaking, the difference between Katok’s shadowing lemma and Proposition 10 is the fact that the compact subsets ${\mathcal{K}_j}$ in Katok’s result are closely related to hyperbolic blocks, while the compact subsets ${\widetilde{\mathcal{O}}}$ in Proposition 10 are holonomy blocks (i.e., subsets of hyperbolic blocks given by certain domination constraints), and hence ${\widetilde{\mathcal{O}}}$ are “usually” smaller than ${\mathcal{K}_j}$. However, from the technical point of view, this difference is not a big deal, but the proof of Proposition 10 from Katok’s shadowing lemma is not exciting (in the sense that we don’t need to introduce new ideas during the process). So, I will proceed as in my lecture and I will skip it (recommending to the curious reader to consult pages 667–671 of Marcelo’s article).

In any event, a direct corollary of Proposition 10 and Proposition 8 is the following result:

Corollary 12 Let ${p_1,\dots,p_{\ell}}$ periodic points as in Proposition 10, and let ${q_i\in W^s_{loc}(p_i)\cap W^u_{loc}(p_\ell)}$, ${i=1,\dots,\ell-1}$. Let ${\xi_a^i\in\mathbb{P}(\mathbb{K}^d)}$, ${1\leq i\leq \ell-1}$, ${1\leq a\leq d}$ such that, for each ${i=1,\dots,\ell-1}$, ${\{\xi_1^i,\dots,\xi_d^i\}}$ are linearly independent. Then,

$\displaystyle \mathcal{U}\ni B\mapsto (h^s_{p_i,q_i}(\xi_a^i))_{\substack{1\leq i\leq \ell-1 \\ 1\leq a\leq d}}\in (\mathbb{P}(\mathbb{K}^d))^{(\ell-1)d}$

is a submersion (where ${\mathcal{U}}$ is a sufficiently small ${C^{r,\nu}}$ neighborhood of ${A}$).

4th step of the proof of Theorem 5: contamination by periodic points (part II)

We start the proof of Theorem 5 with the case ${\mathbb{K}=\mathbb{C}}$. Given ${\ell\in\mathbb{N}}$, we take ${\widetilde{\mathcal{O}}}$ a holonomy block with ${2\ell}$ dominated periodic points ${p_1,\dots,p_{2\ell}\in\widetilde{\mathcal{O}}}$ (with periods ${\kappa_1,\dots,\kappa_{2\ell}}$). By Corollary 9, the map

$\displaystyle \mathcal{U}\ni B\mapsto (B^{\kappa_1}(p_1),\dots, B^{\kappa_{2\ell}}(p_{2\ell}))\in SL(d,\mathbb{C})^{2\ell}$

is a submersion. On the other hand, denoting by ${S\subset SL(d,\mathbb{C})}$ the subset of matrices with a pair of eigenvalues with the same norm. We have that ${S}$ is contained in a finite union of submanifolds of codimension ${\geq 1}$.

Therefore, the set

$\displaystyle \mathcal{Z}_1:=\{B\in\mathcal{U}: B^{\kappa_i}(p_i)\in S \textrm{ for at least } \ell+1 \textrm{ points } p_i\}$

has codimension ${\geq\ell+1}$. Without loss of generality, we can assume that ${B^{\kappa_i}(p_i)\notin S}$ for all ${i=1,\dots,\ell+1}$ and ${B\in\mathcal{U}-\mathcal{Z}_1}$. For each ${i=1,\dots,\ell+1}$, let ${\xi_a^i}$, ${a=1,\dots,d}$, be the eigenspaces of ${B^{\kappa_i}(p_i)}$, and ${q_i\in W^s_{loc}(p_i)\cap W^u_{loc}(p_{\ell+1})}$.

By Corollary 12, the map

$\displaystyle \mathcal{U}\ni B\mapsto (h^s_{p_i,q_i}(\xi_a^i))_{\substack{i=1,\dots,\ell \\ a=1,\dots,d}}\in (\mathbb{P}(\mathbb{K}^d))^{\ell d}$

is a submersion. Hence, the subset

$\displaystyle \mathcal{Z}_2:= \{B\in\mathcal{U}-\mathcal{Z}_1: \forall i=1,\dots,\ell, \exists\, a,b\in\{1,\dots,d\} \textrm{ with } h^s_{p_i,q_i}(\xi_a^i) = h^u_{p_{\ell+1},q_i}(\xi_b^{\ell+1})\}$

has codimension ${\geq\ell}$.

Fix ${B\in\mathcal{U}-(\mathcal{Z}_1\cup\mathcal{Z}_2)}$. We claim that ${\lambda^+(B,x)>0}$ for ${x}$ in subset of positive ${\mu}$-measure set. Indeed, suppose that this is not the case. Then, we will reach a contradiction by the following contamination by periodic points argument.

On one hand, by definition of ${\mathcal{Z}_1}$ and ${\mathcal{Z}_2}$, we have that

$\displaystyle h^s_{p_i,q_i}(\xi_a^i) \neq h^u_{p_{\ell+1},q_i}(\xi_b^{\ell+1}) \ \ \ \ \ (1)$

for all ${i=1,\dots,\ell}$ and ${a,b\in\{1,\dots,d\}}$. On the other hand, by Proposition 7, we know that if ${\lambda^+(B,x)=0}$ for ${\mu}$-a.e. ${x}$, then any ${f_B}$-invariant probability ${m}$ on ${M\times\mathbb{P}(\mathbb{K}^d)}$ (projecting to ${\mu}$) admits a disintegration ${\{\widetilde{m}_z: z\in M\}}$ such that

• the map ${z\mapsto \widetilde{m}_z}$ is continuous,
• ${\widetilde{m}_z}$ is invariant by holonomies.

Now, we observe that if ${m}$ is ${f_B}$-invariant, then ${\widetilde{m}_{f(z)}=B(z)_*\widetilde{m}_z}$ for ${\mu}$-a.e. ${z}$. Hence, by continuity, it follows that

$\displaystyle B^{\kappa_i}(p_i)_*\widetilde{m}_{p_i} = \widetilde{m}_{p_i}$

for all ${i=1,\dots,2\ell}$. Here, we also used that ${p_i\in\textrm{supp}(\mu|_{\widetilde{\mathcal{O}}\cap f^{-\kappa_i}(\widetilde{\mathcal{O}})})}$ (by Proposition 10). In particular, if ${B\notin\mathcal{Z}_1}$, we get that, for each ${i=1,\dots,\ell+1}$, ${\widetilde{m}_{p_i}}$ is a convex combination of Dirac measures supported on ${\xi_a^i}$‘s. Combining this with the invariance by holonomies, we obtain that, for any ${a\in\{1,\dots,d\}}$ such that ${\xi_a^{\ell+1}\in\textrm{supp}(\widetilde{m}_{p_{\ell+1}})}$ and for every ${i\in\{1,\dots,\ell\}}$, one can find ${b\in\{1,\dots,d\}}$ with

$\displaystyle h^s_{p_i,q_i}(\xi_b^i) = h^s_{p_{\ell+1},q_i}(\xi_a^i)$

Of course, this is a contradiction with (1)showing that if ${B\in\mathcal{U}}$ satisfies ${\lambda^+(B,x)=0}$ for ${\mu}$-a.e. ${x}$, then ${B\in\mathcal{Z}_1\cup\mathcal{Z}_2}$. Since ${\mathcal{Z}_1}$ and ${\mathcal{Z}_2}$ have codimension ${\geq\ell}$ and ${\ell\in\mathbb{N}}$ is arbitrary, the proof of Theorem 5 in the case ${\mathbb{K}=\mathbb{C}}$ is complete.

Next, we consider the case ${\mathbb{K}=\mathbb{R}}$. Here, the subset ${S\subset SL(d,\mathbb{R})}$ of matrices with a pair of eigenvalues with the same norm hasn’t codimension ${\geq 1}$: actually, the subset of matrices of ${SL(d,\mathbb{R})}$ with a pair of complex (non-real) conjugated eigenvalues is open! Hence, the argument employed above to deal with the case ${\mathbb{K}=\mathbb{C}}$ can’t be mimicked here.

To overcome this technical difficulty, we recall the following standard trick to “convert” a pair of complex conjugated eigenvalues of a matrix ${A}$ into two real eigenvalues of distinct norms of a matrix ${B}$ close to some power ${A^k}$ of ${A}$: if ${e^{i\alpha}}$ is a (complex) eigenvalue of ${A}$, then ${e^{ik\alpha}}$ is an eigenvalue of ${A^k}$; by appropriately choosing ${k}$, we have that ${e^{ik\alpha}}$ can be made arbitrarily close to ${1}$ (i.e., ${k\alpha}$ close ${0}$ mod ${\mathbb{Z}}$); then, by a preliminary small perturbation of ${A^k}$, we can make ${e^{ik\alpha}}$ and its complex conjugate ${e^{-ik\alpha}}$ both equal to ${1}$, and, by another small perturbation, we can “separate” them into two real eigenvalues of distinct norms (say one with norm ${<1}$ and the other with norm ${>1}$).

The idea in the general setting of ${C^{r,\nu}}$ cocycles ${A:M\rightarrow SL(d,\mathbb{R})}$ over a non-uniformly hyperbolic basis ${(f,\mu)}$ is simply to adapt the previous trick, namely, if some of the periodic points ${p_i}$ considered above (in a holonomy block ${\widetilde{O}}$) is “problematic”, i.e., ${B^{\kappa_i}(p_i)}$ has some pair of complex conjugated eigenvalues, one uses a result of C. Bonatti and M. Viana allowing to replace ${p_i}$ by a nearby periodic point ${\overline{p}_i}$ whose period ${\overline{\kappa}_i}$ is a (well-chosen) multiple of ${\kappa_i}$ so that ${B^{\overline{\kappa}_i}(\overline{p}_i)}$ has a pair of real eigenvalues of distinct norms, up to excluding a subset of ${C^{r,\nu}}$ cocycles of positive codimension. Then, up to performing finitely many of such “exclusions” of positive codimension subsets of ${C^{r,\nu}}$ cocycles, we can assume that the periodic points ${p_1,\dots,p_{2\ell}}$ in the holonomy block ${\widetilde{\mathcal{O}}}$ were chosen so that ${B^{\kappa_i}(p_i)}$ only has real eigenvalues of distinct norms, and thus the same argument of the case ${\mathbb{K}=\mathbb{C}}$ can be applied from this point on. For more details, we recommend reading pages 674-675 of Marcelo’s article.

At this point, our considerations on Theorem 5 of the previous post are finished. Closing this section, let us say a few words on the proofs of Corollaries 6 and 7 of the previous post. In these corollaries, the measure ${\mu}$ was not assumed to be ergodic, but a classical argument based on the local product structure assumption on ${\mu}$ and a Hopf-like argument (see Lemma 5.1 of Marcelo’s article) says that ${\mu}$ has countably many ergodic components ${\mu_j}$, ${j\in\mathbb{N}}$ in the case of non-uniformly hyperbolic systems ${(f,\mu)}$ and ${\mu}$ has finitely many ergodic components ${\mu_j}$ in the case of uniformly hyperbolic homeomorphisms ${(f,\mu)}$, i.e., ${\mu=\sum\limits_{j\in F} a_j\mu_j}$ with ${F\subset\mathbb{N}}$ countable or finite. Moroever, each ${\mu_j}$, ${j\in F}$, has local product structure, and hence Theorem 5 can be applied to each ${(f,\mu_j)}$, ${j\in F}$. As a consequence, we get an open and dense subset ${\mathcal{A}_j}$ of ${\mathcal{S}^{r,\nu}}$ with ${\infty}$-codimension such that, for every ${A\in\mathcal{A}_j}$, it holds ${\lambda^+(A,x)>0}$ for ${\mu_j}$-a.e. ${x}$.

Therefore, the subset ${\mathcal{A}=\bigcap\limits_{j\in F} \mathcal{A}_j}$ of ${\mathcal{S}^{r,\nu}}$ is residual when ${F}$ is countable (the case of ${(f,\nu)}$ non-uniformly hyperbolic) and open, dense with ${\infty}$-codimension when ${F}$ is finite (the case of ${f}$ uniformly hyperbolic). Moreover, any ${A\in\mathcal{A}}$ satisfies ${\lambda^+(A,x)>0}$ for ${\mu}$-a.e. ${x}$. This proves the desired corollaries.

In this section, we collect some natural open questions motivated by Marcelo’s theorem.

During our discussion, we dealt exclusively with the group ${G=SL(d,\mathbb{K})}$. In fact, the main properties we used about this group (see Proposition 8) were:

• Every ${A\in G}$ can be approximated by matrices in ${G}$ with eigenvalues of distinct norms;
• the map ${G\ni B\mapsto (B(\xi_1),\dots,B(\xi_d))\in(\mathbb{P}(\mathbb{K}^d))^d}$ is a submersion whenever ${\xi_1,\dots,\xi_d\in\mathbb{P}(\mathbb{K})^d}$ are linearly independent.

This last property requires ${\textrm{dim}(G)\geq d(d-1)}$. In particular, we “miss” important classical groups in Dynamics, such as the symplectic group ${Sp(d,\mathbb{K})}$ (whose dimension is ${d(d+1)/2}$). So, one natural question is: classify the Lie groups ${G\subset GL(d,\mathbb{K})}$ such that “typical” ${G}$-valued cocycles satisfy the conclusion of Marcelo’s theorem.

Another natural question is about the “topological nature” of the subset ${\mathcal{Z}_0}$ of cocycles with vanishing top exponent. For instance, is ${\mathcal{Z}_0}$ closed? if not, does the closure of ${\mathcal{Z}_0}$ have ${\infty}$-codimension? Of course, the first question is stronger than the second one, and both of them would follow easily if we knew that the Lyapunov exponents ${F\mapsto \lambda_i(F):=\int\lambda_i(F,x) d\mu(x)}$ vary continuously with the cocycle ${F\in\mathcal{S}^{r,\nu}}$ in the ${C^{r,\nu}}$-topology (${r+\nu>0}$). In particular, this motivates the question: when do the Lyapunov exponents vary continuously? To the best of my knowledge, the current scenario is the following: for ${C^0 = C^{0,0}}$-topology, we know (after the works of Bochi and Viana, see here and here) that ${F}$ is a point of ${C^0}$-continuity whenever the Oseledets splitting is dominated or trivial. Moreover, Bochi and Viana have examples of discontinuity for the ${C^{0,\nu}}$-topology with ${\nu>0}$ small. Also, it was recently shown by Bocker and Viana that Lyapunov exponents of locally constant ${GL(2,\mathbb{C})}$ cocycles over Bernoulli shifts vary continuously with the cocycle (and the probability weights).

Closing this section, we recall that Bonatti, Gomez-Mont, Viana showed that the subset ${\mathcal{A}}$ of cocycles with non-vanishing Lyapunov exponents can be chosen independently of the measure ${\mu}$ if we consider exclusively dominated cocycles (in an appropriated sense). Of course, a natural question is whether such a choice of ${\mathcal{A}}$ (independently of ${\mu}$) can be made in general, i.e., in the statement of Theorem 5.

Appendix: proof of Proposition 5

We end today’s post with the proof of Proposition 5 (on the construction of Markov-like partitions). Take ${N\geq 1}$ with ${K e^{-N\tau}<1/4}$ and set ${g=f^N}$. For each ${z\in\mathcal{N}_{\overline{x}}^u(\delta)}$, let ${S_0(z):=[z,\mathcal{N}_{\overline{x}}^s(\delta)]}$ and define inductively

$\displaystyle S_{n+1}(z) = S_n(z) \cup \bigcup\limits_{(j,w)\in Z_n(z)}g^j(S_n(w))$

where ${Z_n(z)=\{(j,w): g^j(S_n(z)) \cap S_0(z)\neq\emptyset\}}$. By definition, ${S_{n+1}(z)\supset S_n(z)}$ and ${Z_{n+1}(z)\supset Z_n(z)}$ for all ${n\geq 0}$.

Definition 13 Let ${S_{\infty}(z):=\bigcup\limits_{n=0}^{\infty} S_n(z)}$ and ${Z_{\infty}(z):=\bigcup\limits_{n=0}^{\infty} Z_n(z)}$, i.e.,

$\displaystyle Z_{\infty}(z) = \{(j,w): g^j(S_{\infty}(w)) \cap S_0(z)\neq\emptyset\}$

and

$\displaystyle S_{\infty}(z) = S_0(z) \cup \bigcup\limits_{(j,w)\in Z_{\infty}(z)} g^j(S_{\infty}(w)).$

Definition 14 Let ${S(z):=S_{\infty}(z)-\bigcup\limits_{(k,\zeta)\in V(z)} g^k(S_{\infty}(\zeta))}$ where ${V(z)=\{(k,\zeta): g^k(S_{\infty}(\zeta))\not\subset S_{\infty}(z)\}}$.

We claim that ${S(z)}$ satisfies the conclusions of Proposition 5. Indeed, we begin by showing that ${S(z)}$ is not very small:

Lemma 15 ${S_0(z)\subset S(z)\subset S_{\infty}(z)\subset W^s_{loc}(z)}$ for all ${z\in\mathcal{N}_{\overline{x}}^u(\delta)}$.

Proof: By definition of ${Z_{\infty}(z)}$ and ${S_{\infty}(z)}$, one has ${g^k(S_{\infty}(\zeta))\cap S_0(z)=\emptyset}$ for all ${(k,\zeta)\in V(z)}$. Hence, ${S_0(z)\subset S(z)\subset S_{\infty}(z)}$. Now, given ${z\in\mathcal{N}_{\overline{x}}^u(\delta)}$ and ${0\leq n\leq \infty}$, let

$\displaystyle \Delta_n = \sup\{\textrm{dist}(z,\eta): \eta\in S_n(z) \textrm{ and } z\in\mathcal{N}_{\overline{x}}^u(\delta)\}.$

We know that ${\Delta_0\rightarrow 0}$ as ${\delta\rightarrow 0}$ (since ${\Delta_0\leq 2\delta}$). So, if ${\delta}$ is sufficiently small, we have ${W^s_{loc}(z)\supset B_{2\Delta_0}(z)}$ for all ${z\in\mathcal{N}_{\overline{x}}^u(\delta)}$. On the other hand, ${\textrm{diam}(g^j(E))\leq (Ke^{-N\tau})^j\textrm{diam}(E)<(1/4^j) \textrm{diam}(E)}$ for all ${E\subset W^s_{loc}(z)}$ and ${j\geq 1}$. Therefore,

$\displaystyle \Delta_{n+1}\leq \Delta_0+\frac{1}{4}\sup\limits_{w\in\mathcal{N}_{\overline{x}}^u(\delta)}\textrm{diam}(S_n(w))\leq \Delta_0 + \frac{1}{2}\Delta_n,$

so that ${\Delta_n\leq 2\Delta_0}$ for all ${n\in\mathbb{N}}$. In particular, it follows that ${S_{\infty}(z)\subset W^s_{loc}(z)}$. This completes the proof of the lemma. $\Box$

Next, we will show that ${S(z)}$ satisfies the Markov-like property. To do so, we need the following lemma:

Lemma 16 Suppose that ${g^{\ell}(S(\zeta))\cap S_{\infty}(z)\neq\emptyset}$. Then,

• (a) ${g^{\ell}(S_{\infty}(\zeta))\subset S_{\infty}(z)}$ and
• (b) for all ${(k,\xi)\in V(z)}$, if ${g^{\ell}(S(\zeta))\cap g^k(S_{\infty}(\xi))\neq\emptyset}$, then ${g^{\ell}(S(\zeta))\subset g^k(S_{\infty}(\xi))}$.

Proof: Let us prove item (a). If ${g^{\ell}(S(\zeta))\subset g^{\ell}(S_{\infty}(\zeta))}$ intersects ${S_0(z)}$, then ${(\ell,\zeta)\in Z_{\infty}(z)}$, and hence ${g^{\ell}(S(\zeta))\subset S_{\infty}(z)}$. Thus, our task is reduced to consider the case ${g^{\ell}(S(\zeta))\cap g^j(S_{\infty}(w))\neq\emptyset}$ for some ${(j,w)\in Z_{\infty}(z)}$.

If ${\ell\leq j}$, one gets ${S(\zeta)\cap g^{j-\ell}(S_{\infty}(w))\neq\emptyset}$. Thus, ${g^{j-\ell}(S_{\infty}(w))\subset S_{\infty}(\zeta)}$ (since, otherwise, ${(j-\ell,w)\in V(\zeta)}$, and hence ${S(\zeta) \cap g^{j-\ell}(S_{\infty}(w))=\emptyset}$). Therefore, ${g^j(S_{\infty}(w))\subset g^{\ell}(S_{\infty}(\zeta))}$. In particular, ${(\ell,\zeta)\in Z_{\infty}(z)}$ (because ${(j,w)\in Z_{\infty}(z)}$ implies ${g^j(S_{\infty}(w))\cap S_0(z)\neq\emptyset}$) and, a fortiori, ${g^{\ell}(S_{\infty}(\zeta))\subset S_{\infty}(z)}$.

The remaining case ${\ell> j}$ is similar: one has ${g^{\ell-j}(S(\zeta)) \cap S_{\infty}(w)\neq\emptyset}$, and so we fit into the hypothesis of the lemma with ${\ell}$ replaced by ${\ell-j<\ell}$ and ${z}$ replaced by ${w}$. Hence, by induction, we complete the proof of item (a) (as the case ${\ell=0}$ is immediate).

Now, let us prove item (b). If ${\ell\leq k}$, we have ${S(\zeta)\cap g^{k-\ell}(S_{\infty}(\xi))\neq\emptyset}$, and hence, by definition of ${S(z)}$, one gets ${g^{k-\ell}(S_{\infty}(\xi))\subset S_{\infty}(\zeta)}$. By item (a) (we just proved), it follows that ${g^k(S_{\infty}(\xi))\subset g^{\ell}(S_{\infty}(\zeta))\subset S_{\infty}(z)}$, a contradiction with the fact that ${(k,\xi)\in V(z)}$. If ${\ell>k}$, we have ${g^{\ell-k}(S(\zeta))\cap S_{\infty}(\xi)\neq\emptyset}$, so that, by item (a),

$\displaystyle g^{\ell-k}(S(\zeta))\subset g^{\ell-k}(S_{\infty}(\zeta))\subset S_{\infty}(\xi).$

Thus, ${g^{\ell}(S(\zeta))\subset g^{k}(S(\xi))}$, and the proof of the lemma is complete. $\Box$

Finally, we end the proof of Proposition 5 by verifying the Markov-like property. Assume that ${g^{\ell}(S(\zeta))\cap S(z)\neq\emptyset}$. By item (a) of the previous lemma, it follows that ${g^{\ell}(S(\zeta))\subset g^{\ell}(S_{\infty}(\zeta))\subset S_{\infty}(z)}$. Therefore, by definition of ${S(z)}$, it suffices to check that ${g^{\ell}(S(\zeta))\cap g^k(S_{\infty}(\xi))=\emptyset}$ for all ${(k,\xi)\in V(z)}$. By item (b) of the previous lemma, if ${g^{\ell}(S(\zeta))\cap g^k(S_{\infty}(\xi))\neq\emptyset}$ for some ${(k,\xi)\in V(z)}$, then ${g^{\ell}(S(\zeta))\subset g^k(S_{\infty}(\xi))}$, and hence, by definition of ${V(z)}$ and ${S(z)}$, ${g^{\ell}(S(\zeta)) \cap S(z)=\emptyset}$, a contradiction. This ends the argument.