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Register a new userNotes on $s$ and $u$-states for cocycles over partially hyperbolic maps
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Authors
Mauricio Poletti
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In these notes we prove that the $s$ or $u$-states of cocycles over partially hyperbolic maps are closed in the space of invariant measures.
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Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by Furstenberg, Guivarch-Raugi, Goldsheid-Margulis and, more recently, Bonatti-Viana and Avila-Viana. In all the cases, the authors consider cocycles over hyperbolic systems, such as shift maps or Axiom A diffeomorphisms. In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of $u$-states. Our main results are stated for certain partially hyperbolic skew-products whose iterates have bounded derivatives along center leaves. They allow us, in particular, to exhibit non-trivial examples of stable simplicity in the partially hyperbolic setting.
Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphsim $f$. We define the u-pressure $P^u(f, varphi)$ of $f$ at a continuous function $varphi$ via the dynamics of $f$ on local unstable leaves. A variational principle for unstable pressure $P^u(f, varphi)$, which states that $P^u(f, varphi)$ is the supremum of the sum of the unstable entropy and the integral of $varphi$ taken over all invariant measures, is obtained. U-equilibrium states at which the supremum in the variational principle attains and their relation to Gibbs u-states are studied. Differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Fr{e}chet differentiability and their relations to u-equilibrium states, are also considered.
We prove a Livv{s}ic-type theorem for Holder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,mu)$ is a non-uniformly hyperbolic system and $A:M to GL(d,mathbb{R}) $ is an $alpha$-H{o}lder continuous map satisfying $ A(f^{n-1}(p))ldots A(p)=text{Id}$ for every $pin text{Fix}(f^n)$ and $nin mathbb{N}$, there exists a measurable map $P:Mto GL(d,mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $mu$-almost every $xin M$. Moreover, we prove that whenever the measure $mu$ has local product structure the transfer map $P$ is $alpha$-H{o}lder continuous in sets with arbitrary large measure.
We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We apply the spectral gap property and the $zeta$-Holder regularity of the disintegration of its physical measure to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $delta$, we show that the physical measure varies continuously with respect to a strong $L^infty$-like norm. Moreover, we prove that for certain interesting classes of perturbations its modulus of continuity is $O(delta^zeta log delta)$.
We prove that in an open and dense set, Symplectic linear cocycles over time one maps of Anosov flows, have positive Lyapunov exponents for SRB measures.
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