Posted by: matheuscmss | October 21, 2011

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

Last spring (more precisely, April-May 2011) I participated in a “Groupe de Travail” organized by Sylvain Crovisier and Jerome Buzzi around the theme “Cocycles over hyperbolic dynamics”. As you can see in this webpage here, after a preparatory talk by F. Ledrappier (on his theorem on vanishing of exponents and determinism of the measures on projective spaces which are invariant under the action of random sequences of matrices), I gave two expository talks (one in April 29 and another in May 20) about Marcelo Viana’s article “Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents”.

My plan is to make the notes I prepared for these expositions available here: today’s post is a slightly expanded version of my notes for the first expository talk, and a future post will correspond to my notes for the second (and final) exposition.

Linear Cocycles

Let {\pi:\mathcal{E}\rightarrow M} be a vector bundle whose fibers are isomorphic to {\mathbb{K}^d} where {d\geq 1} and {\mathbb{K}=\mathbb{R} \textrm{ or } \mathbb{C}}. We say that a vector bundle automorphism {F:\mathcal{E}\rightarrow\mathcal{E}} is a linear cocycle over a transformation {f:M\rightarrow M} if {\pi\circ F=f\circ\pi}.

Example 1 Given {f:M\rightarrow M} and a matrix-valued function {A:M\rightarrow GL(d,\mathbb{K})}, we can form a linear cocycle {F} by considering the (trivial) vector bundle {M\times\mathbb{K}^d} and defining {F(x,v)=(f(x),A(x)v)}.

Example 2 Given a diffeomorphism of a manifold {f:M\rightarrow M}, its derivative {Df:TM\rightarrow TM} is a linear cocycle over {f}. We call {Df} the derivative cocycle.

Let {F:\mathcal{E}\rightarrow\mathcal{E}} be a measurable linear cocycle over an invertible map {f:M\rightarrow M} preserving a probability measure {\mu}. Suppose that {\mathcal{E}} comes equipped with a family {\|.\|} of norms {\|.\|_x} on its fibers {\mathcal{E}_x}, {x\in M}, such that {\log\|F_x^{\pm1}\|} is {\mu}-integrable. Here, {F_x} is the linear map {F_x:\mathcal{E}_x\rightarrow\mathcal{E}_{f(x)}} induced by {F}. In this context, Oseledets theorem implies that, for {\mu}-almost every {x\in M}, we have a splitting

\displaystyle \mathcal{E}_x=E^1_x\oplus\dots\oplus E^k_x, \quad k=k(x)

and a collection of real numbers {\lambda_1(F,x)>\dots>\lambda_k(F,x)} such that

\displaystyle \lim\limits_{n\rightarrow\pm\infty}\frac{1}{n}\log\|F_x^n(v_i)\|=\lambda_i(F,x)

for every {v_i\in E_x^i-\{0\}}. Moreover, the Lyapunov exponents {\lambda_i(F,x)} and the Oseledets subspaces{E_x^i} depend measurably on {x}.

Remark 1 Lyapunov exponents {\lambda_i(F,x)} are constant along {f}-orbits. Therefore, if {\mu} is ergodic, then the Lyapunov exponents are constant ({\mu}-almost everywhere). In this case, we denote by {\lambda_i(F,\mu)} the value of these constants.

In the sequel, we will be interested in the positivity of largest Lyapunov exponent

\displaystyle \lambda^+(F,x):=\lim\limits_{n\rightarrow+\infty}\frac{1}{n}\log\|F_x^n\|=\lambda_1(F,x)

under appropriate smoothness conditions on the linear cocycle {F}. In particular, we will need the following definitions:

Definition 1 Given {r\in\mathbb{N}} and {0\leq\nu\leq 1}, the set {\mathcal{G}^{r,\nu}(f,\mathcal{E})} denotes the set of {C^r} linear cocycles {F} over {f} whose {r}th derivative is {\nu}-Hölder continuous. Here, whenever {r\geq 1}, we assume that the basis {M} and the vector bundle {\pi:\mathcal{E}\rightarrow M} have {C^r}-structures. Moreover, given a Riemannian metric {\langle,\rangle} on {\mathcal{E}}, we denote by {\mathcal{S}^{r,\nu}(f,\mathcal{E})} the subset of {\mathcal{G}^{r,\nu}(f,\mathcal{E})} consisting of linear cocycles {F} verifying {\det F_x=1} for all {x\in M}. Below, we will equip {\mathcal{G}^{r,\nu}(f,\mathcal{E})} and {\mathcal{S}^{r,\nu}(f,\mathcal{E})} with its natural {C^{r+\nu}}-topology.


In his article, Marcelo considers linear cocycles over two classes of hyperbolic systems: uniformly hyperbolic homeomorphisms and non-uniformly hyperbolic diffeomorphisms. Below we explain the main features of these systems.

Given a continuous map {f:M\rightarrow M} of a compact metric space and a point {x\in M}, the stable set of {x} is

\displaystyle W^s(x)=\{y\in M: \textrm{dist}(f^n(y),f^n(x))\rightarrow 0 \textrm{ as } n\rightarrow+\infty\}

and the stable set of size {\varepsilon>0} is

\displaystyle W^s_{\varepsilon}(x)=\{y\in M: \textrm{dist}(f^n(y),f^n(x))\leq\varepsilon, \,\,\forall\, n\geq 0\}.

If in addition {f} is invertible, one can define unstable sets and unstable sets of size {\varepsilon>0} by replacing {f^n} by {f^{-n}} in the previous definitions.

The first class of systems is:

Definition 2 A homeomorphism {f:M\rightarrow M} is uniformly hyperbolic if there are {K,\tau,\varepsilon,\delta>0} such that, for every {x\in M},

  • {\textrm{dist}(f^n(y_1),f^n(y_2))\leq K e^{-\tau n}\textrm{dist}(y_1,y_2)} for all {y_1,y_2\in W^s_{\varepsilon}(x)} and {n\geq 0};
  • {\textrm{dist}(f^{-n}(z_1),f^{-n}(z_2))\leq K e^{-\tau n}\textrm{dist}(z_1,z_2)} for all {z_1,z_2\in W^u_{\varepsilon}(x)} and {n\geq 0};
  • if {\textrm{dist}(x_1,x_2)\leq\delta}, then {\#(W^u_{\varepsilon}(x_1)\cap W^s_{\varepsilon}(x_2))=1}; denoting by {[x_1,x_2]} the unique point in {W^u_{\varepsilon}(x_1)\cap W^s_{\varepsilon}(x_2)}, we also require that {[x_1,x_2]} depends continuously on {x_1} and {x_2}.

The second class of systems is:

Definition 3 Let {f:M\rightarrow M} be a {C^{1+\alpha}}-diffeomorphism ({\alpha>0}) of a compact manifold {M} and {\mu} be a {f}-invariant non-atomic probability. We say that {(f,\mu)} is (non-uniformly) hyperbolic if the Lyapunov exponents {\lambda_i(f,x)=\lambda_i(Df,x)} of the derivative cocycle {Df} are nonzero at {\mu}-almost every {x\in M}.

Given {(f,\mu)} non-uniformly hyperbolic and {x\in M} such that the Lyapunov exponents {\lambda_i(Df,x)} and the Oseledets subspaces {E_x^i} are well-defined, we denote by {E^s_x}, resp. {E^u_x}, the sum of all Oseledets subspaces associated to negative, resp. positive, Lyapunov exponents. Starting from seminal works of Pesin, we dispose nowadays of a whole literature (sometimes called Pesin theory) dedicated to the nice properties of non-uniformly hyperbolic systems. For our purposes, we will need the following properties assured by the so-called Pesin stable manifold theorem: for {\mu}-almost every {x\in M}, there are {W_{loc}^s(x)} and {W_{loc}^u(x)} {C^1}-disks passing through {x} (a.k.a. Pesin local stable and unstable manifolds of {x}) such that

  • at {x}, we have that {W_{loc}^s(x)} is tangent to {E_x^s} and {W_{loc}^u(x)} is tangent to {E_x^u};
  • for every {\tau_x<\min |\lambda_i(x,Df)|}, there exists {K_x>0} such that, for any {n\in\mathbb{N}},
  • (a) {\textrm{dist}(f^n(y_1),f^n(y_2))\leq K_x e^{-n\tau_x}\textrm{dist}(y_1,y_2)} for any {y_1,y_2\in W_{loc}^s(x)};
  •  (b) {\textrm{dist}(f^{-n}(z_1), f^{-n}(z_2))\leq K_xe^{-n\tau_x}\textrm{dist}(z_1,z_2)} for any {z_1,z_2\in W_{loc}^u(x)};
  • {f(W_{loc}^s(x))\subset W_{loc}^s(x)} and {f(W^u_{loc}(x))\supset W_{loc}^u(x)};
  • {W^s(x)=\bigcup\limits_{n=0}^{\infty}f^{-n}(W_{loc}^s(x))} and {W^u(x)=\bigcup\limits_{n=0}^{\infty}f^n(W_{loc}^u(x))}.

Moreover, the constants {\tau_x}, {K_x} and the sizes of the disks {W_{loc}^s(x)} and {W_{loc}^u(x)} can be chosen to depend measurably on {x}.

In a nutshell, Pesin stable manifold theorem says that the measurable plane fields {E^s_x} and {E^u_x} can be locally integrated into local disks {W^s_{loc}(x)} and {W^u_{loc}(x)} whose sizes depend measurably on {x}. Also, the distances of iterates of points in such local disks are exponentially contracted (in the future or in the past) by an exponential rate essentially equal to the Lyapunov exponent of the center {x} of these disks. Finally, any point whose (future or past) iterates converges to the (future or past) iterates of {x} (i.e., any point in the stable {W^s(x)} or unstable {W^u(x)} sets of {x}) must approach it in an exponential way (this is expressed by the fact that, after iterating an adequate number of times, the orbit enters the local disks {W^s_{loc}(x)} or {W^u_{loc}(x)}). For a proof of Pesin stable manifold theorem, we recommend Pesin’s original article or the article of A. Fathi, M. Herman and J.-C. Yoccoz).

The fact that the objects depend measurably on the points allows us the so-called hyperbolic blocks. Roughly speaking, these are “large” compact sets where the objets appearing in Pesin stable manifold theorem depend continuously on the point. More precisely, from the measurable dependence of objects and Luzin theorem, for every {K, \tau>0}, we can select {\mathcal{H}(K,\tau)} a compact set such that

  • {\tau_x\geq\tau} and {K_x\leq K} for any {x\in\mathcal{H}(K,\tau)};
  • the disks {W_{loc}^s(x)} and {W_{loc}^u(x)} depend continuously on {x\in\mathcal{H}(K,\tau)};
  • {\mu(\mathcal{H}(K,\tau))\rightarrow 1} when {\tau\rightarrow 0} and {K\rightarrow\infty}.

In particular, the sizes of {W_{loc}^s(x)} and {W_{loc}^u(x)} and their angles {\angle(E_x^s,E_x^u)} are uniformly bounded away from zero for {x\in\mathcal{H}(K,\tau)}.

Finally, the class of invariant measures {\mu} considered by Marcelo are the measures with a local product structure. In a few words, these are measures whose relationship with the Pesin stable and unstable manifolds is nice. More concretely, given a hyperbolic block {\mathcal{H}(K,\tau)}, we take a small constant {\delta>0} such that, for any two points {x,y\in\mathcal{H}(K,\tau)} with {\textrm{dist}(x,y)\leq\delta}, we have {\#(W_{loc}^s(x)\cap W_{loc}^u(y))=1}. In the sequel, we denote by {[x,y]} the unique point in {W^s_{loc}(x)\cap W_{loc}^u(y)}, and we observe that the corresponding map {(x,y)\mapsto [x,y]} is continuous. Given {x\in\mathcal{H}(K,\tau)}, we define

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

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


\displaystyle \mathcal{N}_x(\delta)=[\mathcal{N}_x^s(\delta),\mathcal{N}_x^u(\delta)],

i.e., {\mathcal{N}_x(\delta)} is the image of {\mathcal{N}_x^s(\delta)\times\mathcal{N}_x^u(\delta)} under the map {[.,.]}. Pictorially, the sets {\mathcal{N}_x^s(\delta)}, {\mathcal{N}_x^u(\delta)} and {\mathcal{N}_x(\delta)} can be seen as follows:

In this context, we have the following definition:

Definition 4 We say that {\mu} has local product structure if, for every {\mu}-generic {x} and {\delta>0} as above, we have that the restriction of {\mu} to {\mathcal{N}_x(\delta)} is equivalent to the product measure {\nu^u\times\nu^s}, where {\nu^u} is the projection of {\mu} to {\mathcal{N}_x^u(\delta)} and {\nu^s} is the projection of {\mu} to {\mathcal{N}_x^s(\delta)}.

Remark 2 This definition makes sense in both cases of non-uniformly hyperbolic diffeomorphisms and uniformly hyperbolic homeomorphisms. In the case of volume-preserving non-uniformly hyperbolic diffeomorphisms {(f,Leb)}, the Lebesgue measure {Leb} has local product structure because, as it was shown by Y. Pesin, the stable and unstable manifolds form absolutely continuous laminations. More generally, every hyperbolic measure {\mu} with absolutely continuous disintegration along the stable and unstable laminations has local product structure (as it was shown by C. Pugh and M. Shub). Finally, any equilibrium measure {\mu} associated to the restriction {f|_{\Lambda}} of an Axiom A {C^{1+\alpha}} diffeomorphism {f} to a basic set {\Lambda} and a Hölder continuous potential {\phi:M\rightarrow\mathbb{R}} (i.e., a probability measure {\mu} verifying the variational principle {h_{\mu}(f)+\int\phi d\mu = \sup\limits_{\nu \, f-\textrm{invariant probability}}h_{\nu}(f)+\int\phi d\nu}) has local product structure (see e.g. R. Bowen’s book).

Statement of results

In the case of {(f,\mu)} non-uniformly hyperbolic, Marcelo showed the following results:

Theorem 5 For all {r\in\mathbb{N}} and {0\leq\nu\leq 1} with {r+\nu>0} and {\mu} ergodic hyperbolic measure with local product structure, the set of cocycles {F\in\mathcal{S}^{r,\nu}(f,\mathcal{E})} whose top Lyapunov exponent is positive, i.e.,

\displaystyle \lambda^+(F,x):=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\log\|A^n(x)\|>0 \quad \textrm{for}\,\mu-\textrm{almost every } x\in M

is open and dense in {\mathcal{S}^{r,\nu}(f,\mathcal{E})}.

Remark 3 Even though I didn’t check all details, I think that the previous statement could be generalized to “non-uniformly hyperbolic homeomorphisms” {(f,\mu)} in the sense that, besides the following Pesin theory like properties

  • for {\mu}-a.e. {x}, there are constants {K_x<\infty}, {\tau_x>0} and “local” stable and unstable disks {W^s_{loc}(x)}, {W^u_{loc}(x)} such that, for any {n\in\mathbb{N}}, \subitem (a) {\textrm{dist}(f^n(y_1),f^n(y_2))\leq K_x e^{-n\tau_x}\textrm{dist}(y_1,y_2)} for any {y_1,y_2\in W_{loc}^s(x)}; \subitem (b) {\textrm{dist}(f^{-n}(z_1), f^{-n}(z_2))\leq K_xe^{-n\tau_x}\textrm{dist}(z_1,z_2)} for any {z_1,z_2\in W_{loc}^u(x)};
  • {K_x}, {\tau_x} and the sizes of the local disks {W^s_{loc}(x)}, {W^u_{loc}(x)} depend measurably on {x};
  • {f(W_{loc}^s(x))\subset W_{loc}^s(x)} and {f(W^u_{loc}(x))\supset W_{loc}^u(x)};
  • {W^s(x)=\bigcup\limits_{n=0}^{\infty}f^{-n}(W_{loc}^s(x))} and {W^u(x)=\bigcup\limits_{n=0}^{\infty}f^n(W_{loc}^u(x))};
  • the local stable and unstable disks are topologically transverse in the sense that, for any hyperbolic block {\mathcal{H}(K,\tau)} (where {K_x\leq K} and {\tau_x\geq\tau}), one can find {\delta>0} such that {\#W^s_{loc}(x)\cap W^u_{loc}(y)=1} whenever {x,y\in\mathcal{H}(K,\tau)} and {\textrm{dist}(x,y)\leq\delta};
  • {\mu} has product local structure in the sense that {\mu|_{\mathcal{N}_x(\delta)}} is equivalent to the product measure {\nu^s\times\nu^u} where {\nu^{s/u}} is the projection of {\mu} to {\mathcal{N}_x^{s/u}(\delta)}, where {\mathcal{N}_x(\delta)}, {\mathcal{N}_x^{s/u}(\delta)} are defined in the same way as above,

one also impose the following “Katok’s shadowing lemma” like property:

  • there is a countable family {\mathcal{K}_m} of “hyperbolic blocks” (say {K_m\subset\mathcal{H}(K_m,\tau_m)}) with {\mu(K_m)\rightarrow 1} as {m\rightarrow\infty} such that, for each {j\in\mathbb{N}} and {\gamma>0}, there are constants {K,\tau,\rho,\varepsilon} such that for every {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} satisfying:
  •  (a) {p\in\mathcal{H}(K,\tau)}
  •  (b) {W^s_{loc}(p)} and {W^u_{loc}(p)} have size {\rho} (at least) and they are topologically transverse to the local stable and unstable disks of any {w\in\mathcal{K}_j} in a {\rho}-neighborhood of {p};
  •  (c) {\textrm{dist}(f^{i}(p),f^{i}(z))<\gamma} for every {0\leq i\leq \kappa}.

However, this notion of “non-uniformly hyperbolic homeomorphisms” is not very useful as the main source of examples of such systems are precisely non-uniformly hyperbolic diffeomorphisms.

Essentially by combining Theorem 5 with the ergodic decomposition theorem, one is able to derive the following corollary:

Corollary 6 For every {r\in\mathbb{N}}, {0\leq \nu\leq 1} with {r+\nu>0}, and {\mu} hyperbolic measure (not necessarily ergodic) with local product structure, the set {\mathcal{A}} of cocycles {F} with {\lambda^+(F,x)>0} for {\mu}-a.e. {x} is a Baire residual subset of {\mathcal{S}^{r,\nu}(f,\mathcal{E})}.

On the other hand, in the case of uniformly hyperbolic homeomorphisms, Marcelo is able to recover the full conclusion of Theorem 5 even for non-ergodic measures:

Corollary 7 For every {r\in\mathbb{N}}, {0\leq \nu\leq 1} with {r+\nu>0}, {f} uniformly hyperbolic homeomorphism, and {\mu} probabilitiy measure with local product structure, the set {\mathcal{A}} of cocycles {F} with {\lambda^+(F,x)>0} for {\mu}-a.e. {x} is open, dense and its codimension is {\infty} in {\mathcal{S}^{r,\nu}(f,\mathcal{E})}.

Remark 4 In Corollary 7, if one assumes that the cocycle {F} is dominated (roughly speaking this means that the dynamics on the fibers has expansion/contraction rates situated “between” the expansion/contraction rates of the base dynamics {f}), then the set {\mathcal{A}} may be chosen independently of measure {\mu} (as it was shown by C. Bonatti, X. Gomez-Mont and M. Viana), and the Lyapunov spectrum is simple, i.e., the multiplicity of all Lyapunov exponents of {F\in\mathcal{A}} is 1 (as it was shown by C. Bonatti and M. Viana).

Partly motivated by the results mentioned in the remark above, Marcelo conjectures that:

Conjecture. Theorem 5 and Corollaries 6, 7 remain true if one replaces “{\lambda^+(F,x)>0}” by “simple Lyapunov spectrum” in their conclusions.

Remark 5 (Historical remark) These results in the same lines of previous theorems of H. Furstenberg, A. Raugi, Y. Guivarc’h, I. Goldsheid, G. Margulis on the Lyapunov spectrum of identically and independently distributed (i.i.d. for short) random products of matrices, and Bonatti, Gomez-Mont, Viana, Bonatti, Viana on the Lyapunov spectrum of uniformly hyperbolic homeomorphisms, and Avila, Viana on the Lyapunov spectrum of the so-called Kontsevich-Zorich cocycle.

Remark 6 The assumption {r+\nu>0} is necessary:

  • by the works of J. Bochi, and J. Bochi, M. Viana, we know that vanishing exponents may be locally {C^0} generic (i.e., one can construct {C^0} open sets of cocycles where vanishing exponents is a {C^0} Baire residual property);
  • by the works of L. Arnold, N. D. Cong and A. Arbieto, J. Bochi, we know that vanishing exponents is a {L^p} generic property for all {1\leq p<\infty}.

Our long-term goal here is to present the proof of Theorem 5. In the next section, we describe some of the main steps towards this result.

Strategy of proof of Theorem 5

We begin with some (technical) preliminary reductions. Firstly, we notice that, up to replacing the metric {d(x,y)} by {d(x,y)^{\nu}} ({\nu>0}), one can assume that our cocycles are Lipschitz, i.e., {\nu=1} and {F\in \mathcal{S}^{r,1}(f,\mathcal{E})}, {r\geq 0}. Secondly, as we’re going to see, all subsequent arguments will be local in nature, so that one can also assume that {\mathcal{E} = M\times\mathbb{K}^d} (where {\mathbb{K} = \mathbb{C}} or {\mathbb{R}}). In particular, under this assumption, we can think of {\mathcal{S}^{r,\nu}(f,\mathcal{E})} as the set

\displaystyle \left\{A:M\rightarrow SL(d,\mathbb{K}): A\in C^{r,\nu}\right\}

equipped with the norm

\displaystyle \|A\|_{r,\nu} = \max\limits_{0\leq i\leq r}\sup\limits_{x\in M} \|D^i A(x)\| + \sup\limits_{x\neq y}\frac{\|D^rA(x)-D^rA(y)\|}{d(x,y)^{\nu}}

In the sequel, an important role will be played by the projective cocycle {f_A: M\times\mathbb{P}(\mathbb{K}^d)\rightarrow M\times\mathbb{P}(\mathbb{K}^d)} naturally associated to a linear cocycle {(f,A)}.

In this language, the proof of Theorem 5 can be divided into the following steps:

  • First step: One shows that {\lambda^+(A,x)=0} for {\mu}-a.e. {x} implies that the cocycle is dominated at {x} (in a sense that is slightly weaker than the one studied by Bonatti, Gomez-Mont, Viana). Then, one shows that this domination at {\mu}-a.e. {x} implies the existence of nice stable and unstable manifolds for the projective cocycle {f_A} at {(x,\xi)\in\{x\}\times\mathbb{P}(\mathbb{K}^d)} (and, moreover, these {f_A}-invariant manifolds are graphs over the stable and unstable manifolds for {f} at {x}). In particular, these {f_A}-invariant manifolds can be used to define stable holonomies {h_{x,y}^s:\{x\}\times\mathbb{P}(\mathbb{K}^d)\rightarrow \{y\}\times\mathbb{P}(\mathbb{K}^d)} for two points {x,y\in M} in the same stable manifold and unstable holonomies {h_{x,z}^u: \{x\}\times\mathbb{P}(\mathbb{K}^d)\rightarrow \{z\}\times\mathbb{P}(\mathbb{K}^d)} for two points {x,z\in M} in the same unstable manifold.
  • Second step: By the compactness of the projective space {\mathbb{P}(\mathbb{K}^d)}, we have that {M\times\mathbb{P}(\mathbb{K}^d)} always supports {f_A}-invariant measures {m} projecting to {\mu} under the natural projection {M\times\mathbb{P}(\mathbb{K}^d)\rightarrow M}. By Rokhlin’s disintegration theorem (see this link here for Rokhlin’s original article, and this link herefor a modern exposition by Marcelo of the same result), any such probability measure {m} can be disintegrated into an essentially unique family {\{m_x: x\in M\}}, that is, we can write

    \displaystyle m(A)=\int_M m_x(A\cap(\{x\}\times\mathbb{P}(\mathbb{K}^d)))d\mu(x)

    and the family is unique in the sense that any two families verifying the previous equation must coincide up to a set of zero {\mu}-measure. By a result of F. Ledrappier, we will see that {\lambda^+(A,x)=0} implies {m_y = (h_{x,y}^s)_*m_x} and {m_z = (h_{x,z}^u)_*m_x}. In other words, the disintegration {\{m_x:x\in M\}} is invariant under stable and unstable holomies whenever {\lambda^+(A,x)=0}. Actually, as I already mentioned, F. Ledrappier gave a talk on his result at the “groupe de travail” as a “preparation” for my expositions. So, during my talks it was assumed previous knowledge of F. Ledrappier’s theorem, and I will also do so here: we’ll content ourselves to state and use Ledrappier’s theorem without further mentions to its proof (even though this is a very interesting theorem lying at the heart of this proof). For more details, I recommend reading the original article (since it is not very long). In any case, the invariance under holonomies of the disintegration of {m} can be used to show that the map {x\rightarrow m_x} is continuous. In particular, as it is known that periodic points are dense when {(f,\mu)} is non-uniformly hyperbolic, this will allow us to say that the dynamical behavior of periodic points affects the entire dynamics. Notice that this “contamination by periodic points” (as Marcelo likes to call it) is only possible when {\lambda^+(A,x)=0}, and it is quite remarkable: even though the set of periodic points has zero {\mu}-measure (as {\mu} is non-atomic), the “non-wildness” of the cocycle (expressed by the property {\lambda^+(A,x)=0}) permits to say that they matter for the global dynamics. Of course, this is a particularity of linear cocycles with vanishing exponents and it is far from being true in general.

  • Third step: Using certain nice properties of the so-called blocks of dominations (analogs of Pesin’s hyperbolic blocks for {f_A}), one can construct an arbitrarily large number of periodic points, all of them being dynamically related (“heteroclinically linked by their invariant manifolds”). Here, it is crucial that {\mu} has local product structure!
  • Fourth step: In the case {\mathbb{K}=\mathbb{C}}, we will complete the proof of Theorem 5 by means of the following argument. Given {\ell\in\mathbb{N}}, we select {p_1,\dots,p_{2\ell}} pairwise distinct periodic points of {f}. Recall that a matrix {A} of {SL(d,\mathbb{C})} having some eigenvalues with the same norm is a phenomenon of codimension 1 (at least). Hence, we have that the fact that the cocycle {A} is “typical” over {p_1,\dots,p_{2l}}, i.e., the eigenvalues of {A^{\kappa(p_j)}(p_j)} has eigenvalues of distinct norms (where {\kappa(p_j)} is the {f}-period of {p_j}) for each {j=1,\dots,2\ell} has codimension {\geq 2\ell\geq\ell}. On the other hand, if {\lambda^+(A,x)=0} at {\mu}-a.e. {x}, by the second step above we have that {(h_{p_j,q}^u)_* m_{p_j} = m_q = (h_{p_k,q}^s)_* m_{p_k}} for all {q\in W^u(p_j)\cap W^s(p_k)}. Moreover, since the cocycle {A} is typical over {p_1,\dots,p_{2\ell}}, we know that {m_{p_j}} is a linear combination of Dirac measures supported on the eigenspaces of {A^{\kappa(p_j)}(p_j)}. Hence, the equality {(h_{p_j,q}^u)_* m_{p_j} = (h_{p_k,q}^s)_* m_{p_k}} implies that the {h_{p_j,q}^u}-image of some eigenspace of {A^{\kappa(p_j)}(p_j)} coincides with the {h_{p_k,q}^s}-image of some eigenspace of {A^{\kappa(p_k)}(p_k)}. As we’re going to see later, this coincidence at one heteroclinic point {q\in W^u(p_j)\cap W^s(p_k)} is a positive codimension phenomenon, so that its validity at all heteroclinic points is a codimension {\geq\ell} phenomenon. Because {\ell\in\mathbb{N}} is an arbitrary integer, we see that set of cocycles {A\in SL(d,\mathbb{C})} with {\lambda^+(A,x)=0} at {\mu}-a.e. {x} has {\infty} codimension, that is, one gets Theorem 5 in the case {\mathbb{K}=\mathbb{C}}. However, in the remaining case {\mathbb{K}=\mathbb{R}}, we can not proceed as above: indeed, the set of matrices in {SL(d,\mathbb{R})} with a pair of complex conugate eigenvalues is open, so that we can’t say anymore that “a matrix with some eigenvalues of same norm is a codimension 1 phenomenon”. In particular, this case will introduce a few technical issues that we prefer comment only in due time.

This being said, let me mention that we’ll split the proof of Theorem 5 into two blog posts (in a way more or less corresponding to my two talks at the “groupe de travail”). More precisely, in the remainder of today’s post we will give more details on the first step above, and we will leave the discussion of the other three steps for a subsequent post.

1st step of the proof of Theorem 5: domination and invariant foliations

We start the final section of today’s post with the notion of “domination”. Given {(f,\mu)} a non-uniformly hyperbolic system, {A:M\rightarrow SL(d,\mathbb{K})} a {C^{r,\nu}}-cocycle and {\mathcal{H}(K,\tau)} a hyperbolic block of {(f,\mu)}, we define, for each {N\in\mathbb{N}}, {\theta>0}, {D_A(N,\theta)} as the set of points {x\in M} such that

\displaystyle \prod\limits_{j=0}^{k-1}\|A^N(f^{jN}(x))\|\cdot\|A^N(f^{jN}(x))^{-1}\|\leq e^{kN\theta}


\displaystyle \prod\limits_{j=0}^{k-1}\|A^{-N}(f^{-jN}(x))\|\cdot\|A^{-N}(f^{-jN}(x))^{-1}\|\leq e^{kN\theta}

for all {k\geq 1} ({k\in\mathbb{N}}).

Definition 8 We say that {x} is {s}-dominated ({s\geq 1}) if {x\in\mathcal{H}(K,\tau)\cap D_A(N,\theta)} where {s\theta<\tau}.

Roughly speaking, since the parameters {K} and {\tau} controls the rates of hyperbolicity of {f} at a point {x\in\mathcal{H}(K,\tau)}, we see that the domination condition says that the “strength of hyperbolicity” (measure by {\theta}) of the cocycle {A} along the fibers {\mathbb{K}^d} can’t surpass the strength of hyperbolicity on the basis {f} (measured by {s} and {\tau}): this is the content of the condition {s\theta<\tau}. In other words, if we consider the dynamics of the projective cocycle {f_A}, then the domination condition {s\theta<\tau} is some sort of quantitative partial hyperbolicity of {f_A} at {x}: the stable and unstable directions of {f_A} correspond to the ones of {f}, while the “central / dominated” direction is the fiber direction {\{0\}\times\mathbb{K}^d} (as, given a matrix {B}, the norm of its projective action {B_{\#}} and its inverse {B_{\#}^{-1}} are bounded from above by {\|B\|\cdot\|B^{-1}\|}; in particular, this justifies the choice of the expressions {\prod\limits_{j=0}^{k-1}\|A^N(f^{jN}(x))\|\cdot\|A^N(f^{jN}(x))^{-1}\|} and {\prod\limits_{j=0}^{k-1}\|A^{-N}(f^{-jN}(x))\|\cdot\|A^{-N}(f^{-jN}(x))^{-1}\|} in the previous definition to measure the strength of hyperbolicity of [iterates of] {f_A} at the fiber directions).

In the sequel, we will to study the relationship between vanishing exponents and domination.

Proposition 9 For {\mu}-a.e. {x} with {\lambda^+(A,x)=0}, we have that {x} is {s}-dominated for every {s\geq 1}.

In few words, this proposition says that the vanishing of Lyapunov exponents at {\mu}-typical point implies “{\infty}-domination” (i.e., {s}-domination for all {s\geq 1}). To prove this proposition, we will need the following Lemma.

Lemma 10 For all {\delta>0} and {\mu}-a.e. {x\in M}, there exists {N=N(x)\geq 1} such that

\displaystyle \frac{1}{k}\sum\limits_{j=0}^{k-1}\frac{1}{N}\log\|A^N(f^{jN}(x))\|\leq\lambda^+(A,x)+\delta

for every {k\geq 1}.

Proof: Take {\varepsilon>0} such that {4\varepsilon\sup\limits_{z\in M}\log\|A(z)\|<\delta} and {\eta\geq 1} a large integer with

\displaystyle \mu(\Delta_{\eta})\geq 1-\varepsilon^2

where {\Delta_{\eta} = \left\{x\in M: \frac{1}{\eta}\log\|A^{\eta}(x)\|\leq \lambda^+(A,x)+\frac{\delta}{2}\right\}}. Let {\tau(x)} be the “average sojourn time” of the {f^{\eta}}-orbit of {x} inside {\Delta_{\eta}} and put {\Gamma_{\eta}=\{x\in M: \tau(x)\geq 1-\varepsilon\}}. By sub-multiplicativity of norms, we have that

\displaystyle \frac{1}{k}\sum\limits_{j=0}^{k-1}\frac{1}{\ell\eta}\log\|A^{\ell\eta}(f^{j\ell\eta}(x))\|\leq \frac{1}{k\ell}\sum\limits_{j=0}^{k\ell-1}\frac{1}{\eta}\log\|A^{\eta}(f^{j\eta}(x))\|

for any {\ell\geq 1}. For {x\in\Gamma_{\eta}}, fix {\ell=\ell(x)} large so that

\displaystyle \#\{j\in\{0,\dots,n-1\}:f^{j\eta}(x)\notin\Gamma_{\eta}\}\leq (1-\tau(x)+\varepsilon)n

for each {n\geq\ell}. It follows that

\displaystyle \begin{array}{rcl} \frac{1}{k\ell}\sum\limits_{j=0}^{k\ell-1}\frac{1}{\eta}\log\|A^{\eta}(f^{j\eta}(x))\| &\leq& \lambda^+(A,x)+\frac{\delta}{2} + (1-\tau(x)+\varepsilon)\sup\log\|A\| \\ &<& \lambda^+(A,x)+\delta \end{array}

By putting the previous two inequalities together, we see that {\mu}-a.e. {x\in\Gamma_{\eta}} satisfy the conclusion of the Lemma with {N=\ell\eta=N(x)}. On the other hand,

\displaystyle \mu(\Gamma_{\eta}) + (1-\varepsilon) \mu(M-\Gamma_{\eta}) \geq \int \tau(x)d\mu = \mu(\Delta_{\eta})\geq 1-\varepsilon^2,

so that {\mu(\Gamma_{\eta})\geq 1-\varepsilon}. Since {\varepsilon>0} is arbitrary, by letting {\varepsilon\rightarrow 0}, we see that the proof of the Lemma is complete. \Box

Corollary 11 Let {\theta>0} and {\lambda\geq 0} with {d\cdot\lambda<\theta} (where {d} is the dimension of the fiber {\mathbb{K}^d}). Then, for {\mu}-a.e. {x} with {\lambda^+(A,x)\leq\lambda}, one has {x\in D_A(N,\theta)} for some {N\geq 1}.

Proof: Take {\delta>0} with {d(\lambda+\delta)<\theta}, and {x\in M}, {N=N(x)\geq 1} satisfying the conclusion of the previous Lemma, i.e.,

\displaystyle \frac{1}{k}\sum\limits_{j=0}^{k-1}\frac{1}{N}\log\|A^N(f^{jN}(x))\|\leq\lambda^+(A,x)+\delta

Since {A\in SL(d,\mathbb{K})}, we have that {\det A^N(z)=1} for all {z\in M}, and, a fortiori, {\|A^N(z)^{-1}\|\leq\|A^N(z)\|^{d-1}} for all {z\in M}. Hence,

\displaystyle \frac{1}{kN}\sum\limits_{j=0}^{k-1}\log\left(\|A^N(f^{jN}(x))\|\cdot\|A^N(f^{jN}(x))^{-1}\|\right) \leq d(\lambda^+(A,x)+\delta) < \theta,

that is, {x\in D_A(N,\theta)}. \Box

At this stage, it is not hard to check that Proposition 9 is a direct consequence of this corollary.

Remark 7 One of the reasons that Marcelo treats the issue of vanishing top Lyapunov exponent {\lambda^+(A,x)} but not the simplicity one is more or less explained by the proof of Proposition 9: while absence of positive top exponent (i.e., {\lambda^+=0}) implies a certain “domination” (a crucial ingredient in the proof of Theorem 5 as it allows the existence of holonomies and contamination by periodic points), it is not obvious that the absence of simplicity implies some sort of domination or other nice property allowing for the “contamination by periodic points” argument. In fact, as the reader can see from these two articles of C. Bonatti, M. Viana, and A. Avila, M. Viana, in all situations (as far as I know) where one can deduce simplicity, either the cocycle is (a priori) assumed to be dominated (the case of Bonatti-Viana article) or locally constant (the case Avila-Viana article). In the former case, the existence of holonomies follows the arguments we present below, while in the former case, the existence of holonomies is granted for free (even in absence of domination).

Our next goal is to derive the existence of nice strong stable and unstable manifolds of the projective cocycle {f_A} at {(x,\xi)} (which are Lipschitz graphs over the stable and unstable manifolds of {f} at {x}) whenever {x} is {2}-dominated. To do so, we need a preliminary result about the existence of holonomies and {1}-dominated points.

Proposition 12 There exists {L>0} such that, for every {x} {1}-dominated, say, {x\in\mathcal{H}(K,\tau)\cap D_A(N,\theta)} with {\theta<\tau}, and {y,z\in W^s_{loc}(x)}, the following limit

\displaystyle H_{y,z}^s:=H_{A,y,z}^s:=\lim\limits_{n\rightarrow+\infty} A^n(z)^{-1} A^n(y)

(the stable holonomy between {y} and {z}) exists, and, moreover,

\displaystyle \|H_{y,z}^s-Id\|\leq L\textrm{dist}(y,z)

and {H_{y,z}^s = H_{x,z}^s\circ H_{y,x}^s}.

The proof of this proposition relies on the following “bounded distortion” type lemma:

Lemma 13 There exists a constant {C=C(A,K,\tau,N)>0} such that

\displaystyle \|A^n(y)\|\cdot\|A^n(z)^{-1}\|\leq C e^{n\theta}

for all {y,z\in W^s_{loc}(x)}, {x\in D_A(N,\theta)}.

Proof: Firstly, we observe that

\displaystyle \|A^n(y)\|\cdot\|A^n(z)^{-1}\|\leq C_1\prod\limits_{j=0}^{k-1}\|A^N(f^{jN}(y))\|\cdot\|A^N(f^{jN}(z))^{-1}\|

where {k=\lfloor n/N\rfloor} and {C_1=C_1(A,N)}.

Secondly, since {A} being a Lipschitz cocycle, one has, for some constant {L_1 = L_1(A,N)},

\displaystyle \frac{\|A^N(f^{jN}(y))\|}{\|A^N(f^{jN}(x))\|}\leq \exp(L_1 \textrm{dist}(f^{jN}(x),f^{jN}(y))\leq \exp(L_1\cdot K\cdot e^{-jN\tau})


\displaystyle \frac{\|A^N(f^{jN}(z))^{-1}\|}{\|A^N(f^{jN}(x))^{-1}\|}\leq \exp(L_1\textrm{dist}(f^{jN}(x),f^{jN}(z)))\leq \exp(L_1\cdot K\cdot e^{-jN\tau})

whenever {y,z\in W^s_{loc}(x)}.

Finally, by the domination assumption {x\in D_A(N,\theta)}, we have

\displaystyle \prod\limits_{j=0}^{k-1}\|A^N(f^{jN}(x))\|\cdot\|A^N(f^{jN}(x))^{-1}\|\leq e^{kN\theta}\leq e^{n\theta}

By putting these estimates together, one can check that the Lemma follows (with {C = C_1\exp(L_1\cdot K\cdot\sum\limits_{j=0}^{\infty}e^{-jN\tau}}). \Box

Now we can complete the proof of Proposition 12:

Proof: We claim that {A^n(z)^{-1}A^n(y)} is a Cauchy sequence. Indeed, we observe that

\displaystyle \|A^{n+1}(z)^{-1}A^{n+1}(y) - A^n(z)^{-1}A^n(y)\|\leq

\displaystyle \|A^n(z)^{-1}\|\cdot\|A(f^n(z))^{-1}A(f^n(y))-Id\|\cdot\|A^n(y)\|

Since {A} is Lipschitz,

\displaystyle \|A(f^n(z))^{-1}A(f^n(y))-Id\|\leq L_2\textrm{dist}(f^n(y),f^n(z))\leq L_2 K e^{-n\tau}\textrm{dist}(y,z)

whenever {y,z\in W^s_{loc}(x)}. By combining these estimates with Lemma 13, one obtains

\displaystyle \|A^{n+1}(z)^{-1}A^{n+1}(y) - A^n(z)^{-1}A^n(y)\|\leq C L_2 e^{n(\theta-\tau)}\textrm{dist}(y,z) \ \ \ \ \ (1)

Because {\theta<\tau} (by {1}-domination), the claim is proved. In particular, the limit {H_{y,z}^s:=\lim\limits_{n\rightarrow+\infty} A^n(z)^{-1} A^n(y)} exists and it satisfies

\displaystyle \|H_{y,z}^s-Id\|\leq L \textrm{dist}(y,z)

with {L=CL_2\sum\limits_{n=0}^{\infty}e^{n(\theta-\tau)}}. Finally, the verification of the identity {H_{y,z}^s = H_{x,z}^s\circ H_{y,x}^s} is left as an exercise to the reader. \Box

Corollary 14 There exists {\widetilde{L}>0} such that for every {2}-dominated {x}, say {x\in\mathcal{H}(K,\tau)\cap D_A(N,\theta)}, {2\theta<\tau}, and {y,z\in W^s_{loc}(x)}, the following limit

\displaystyle H_{f^j(y),f^j(x)}^s:=\lim\limits_{n\rightarrow\infty} A^n(f^j(z))^{-1} A^n(f^j(y)) = A^j(z)\cdot H_{y,z}^s\cdot A^j(y)^{-1}

exists and it satisfies

\displaystyle \|H_{f^j(y),f^j(x)}^s-Id\|\leq\widetilde{L} e^{j(2\theta-\tau)}\textrm{dist}(y,z)\leq\widetilde{L}\textrm{dist}(y,z)

Proof: Since {A^n(f^j(z))^{-1} A^n(f^j(y)) = A^j(z) [A^{n+j}(z)^{-1}A^{n+j}(y)] A^j(y)^{-1}}, the desired limit exists (by Proposition 12) and it is {A^j(z)\cdot H_{y,z}^s\cdot A^j(y)^{-1}}. Moreover, by the bounded distortion type Lemma 13 and the estimate (1) above (with {n} replaced by {n+j}), one obtains

\displaystyle \|A^{n+1}(f^j(z))^{-1} A^{n+1}(f^j(y))-A^n(f^j(z))^{-1} A^n(f^j(y))\|\leq .

C e^{j\theta} C L_2 K e^{(n+j)(\theta-\tau)}\textrm{dist}(y,z).

By summing over {n\in\mathbb{N}}, we deduce the last statement of the Corollary. \Box

Remark 8 Note that if {x} is dominated for {A}, say {x\in D_A(N,\theta)}, then {x} is dominated for {B} whenever {B} is sufficiently {C^0} close to {A}: more precisely, for each {\theta'>\theta}, we can select a {C^0} neighborhood {\mathcal{U}} of {A} such that {x\in D_B(N,\theta')} when {B\in\mathcal{U}}. Similarly, the reader can check that the constants {L_1, L_2, L} and {\widetilde{L}} above can be taken uniform in a {C^0} neighborhood of {A}. In particular, all statements above hold uniformly in a {C^0} neighborhood of {A}.

Closing today’s post, we study the dependence of the holonomies on the cocycle {A}. In this direction, we get the following result under a {3}-domination assumption.

Lemma 15 Assume that {x} is {3}-dominated for {A}, say {x\in\mathcal{H}(K,\tau)\cap D_A(N,\theta)}, {3\theta<\tau}. Then, there is a {C^{r,\nu}} neighborhood {\mathcal{U}} of {A} such that, for every {y,z\in W^s_{loc}(x)}, the map

\displaystyle B\in\mathcal{U}\mapsto H_{B,y,z}^s

is {C^1} and its derivative is

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

\displaystyle [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]

\displaystyle B^i(y)

Proof: Fix {\theta'>\theta} with {3\theta'<\tau}. By the previous remark, we can select a {C^{r,\nu}} neighborhood {\mathcal{U}} of {A} such that {x\in\mathcal{H}(K,\tau)\cap D_B(N,\theta')} for every {B\in\mathcal{V}}. By the bounded distortion type Lemma 13, {\|B^i(z)^{-1}\|\cdot\|B^i(y)\|\leq C e^{i\theta'}}, and by the previous corollary, {\|H_{B,f^i(y),f^i(z)}^s-Id\|\leq \widetilde{L} e^{i(2\theta'-\tau)}\textrm{dist}(y,z)}. On the other hand, {\|B(f^i(y))^{-1} \dot{B}(f^i(y))\|\leq \|B^{-1}\|_{r,\nu}\|\dot{B}\|_{r,\nu}} and

\|B(f^i(y))^{-1} \dot{B}(f^i(y)) - B(f^i(z))^{-1} \dot{B}(f^i(z))\|\leq

2 L_3 \|\dot{B}\|_{r,\nu} \textrm{dist}(f^i(y),f^i(z))\leq

\displaystyle 2L_3 K e^{-i\tau}\|\dot{B}\|_{r,\nu}\textrm{dist}(y,z)

where {L_3=\sup\{\|B\|_{r,\nu}:B\in\mathcal{U}\}}. It follows that the expression above defining {\partial_B H^s_{B,y,z}(\dot{B})} converges as

\displaystyle \begin{array}{rcl} \|\partial_B H^s_{B,y,z}(\dot{B})\| &\leq& \sum\limits_{i=0}^{\infty}Ce^{i\theta'}2L_3[\widetilde{L}e^{i(2\theta'-\tau)} + Ke^{-i\tau}]\|\dot{B}\|_{r,\nu}\textrm{dist}(y,z) \\ &\leq& \widetilde{C} e^{i(3\theta'-\tau)}\|\dot{B}\|_{r,\nu}\textrm{dist}(y,z) \end{array}

where {\widetilde{C} = 2L_3C(\widetilde{L}+K)}.

Now, we recall that {H^n_{B,y,z}:=B^n(z)^{-1}B^n(y)\rightarrow H^s_{B,y,z}} as {n\rightarrow\infty} and each {H^n_{B,y,z}} is {C^1} on the {B}-variable with derivative

\displaystyle \begin{array}{rcl} \partial_B H_{B,y,z}^n (\dot{B}) &=& B^n(z)^{-1}\sum\limits_{i=0}^{n-1}B^{n-i}(f^i(y))B(f^i(y))^{-1}\dot{B}(f^i(y)) B^i(y) \\ &-& \sum\limits_{i=0}^{n-1} B^i(z)^{-1} B(f^i(z))^{-1} \dot{B}(f^i(z)) B^{n-i}(f^i(z))^{-1} B^n(y) \end{array}

Thus, our task is reduced to show that {\partial_B H_{B,y,z}^n (\dot{B})\rightarrow \partial_B H_{B,y,z}^s (\dot{B})} uniformly. Keeping this goal in mind, we observe that the previous corollary implies that

\displaystyle \|H^{n-i}_{B,f^i(y),f^i(z)} - H^s_{B,f^i(y),f^i(z)}\|\leq \widetilde{L} e^{i\theta'}e^{n(\theta'-\tau)} \textrm{dist}(y,z)

for each {0\leq i\leq n-1}. Thus, the difference between the ith terms of {\partial_B H^n_{B,y,z}(\dot{B})} and {\partial_B H^s_{B,y,z}(\dot{B})} is bounded by

\displaystyle 2Ce^{i\theta'}\widetilde{L}e^{i\theta'}e^{n(\theta'-\tau)}L_3\|\dot{B}\|_{r,\nu}\textrm{dist}(y,z) = \widehat{C} e^{2i\theta'}e^{n(\theta'-\tau)}\|\dot{B}\|_{r,\nu}\textrm{dist}(y,z)

where {\widehat{C} = 2C\widetilde{L}L_3}. Putting this estimate together with the bounds of the previous paragraph applied to the terms {i\geq n}, we deduce that

\displaystyle \|\partial_B H^s_{B,y,z}(\dot{B})-\partial_B H^n_{B,y,z}(\dot{B})\|\leq \left(\widehat{C}\sum\limits_{i=0}^{n-1}e^{2i\theta'}e^{n(\theta'-\tau)} + \widetilde{C} \sum\limits_{i=n}^{\infty} e^{i(3\theta'-\tau)}\right)

\displaystyle \|\dot{B}\|_{r,\nu}\textrm{dist}(y,z).

Because {3\theta'<\tau}, we get that the right-hand side of this estimate goes to {0} as {n\rightarrow\infty}. This proves the Lemma. \Box

So, this is all for today! Next time, we will complete the proof of Theorem 5 by discussing the remaining steps in the strategy of proof presented above.

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