We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.
In this paper we give an upper bound for the number of SRB measures of saddle type of local diffeomorphisms of boundaryless manifolds in terms of maximal cardinality of set of periodic points without any homoclinic relation.
In this paper we improve the results of \cite{MT} and show that a weak hyperbolic transitivity implies the uniqueness of hyperbolic SRB measures. As an important corollary, it arises the ergodicity of the system in a conservative setting. It also arises the condition which implies the stable ergodicity as well as the statistical stability for a general $C^2$-regular map.
In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar diffeomorphisms have contraction on average which forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one $SRB$ measure. In our approach, these skew product systems arising from iterated function systems generated by a finitely many weak contractive diffeomorphisms. Under some conditions including …
In the present paper we contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures…
We construct SRB measures for endomorphisms satisfying conditions far weaker than the non-uniformly expansion. As a consequence, the definition of non-uniformly expanding map can be weakened. We also prove the existence of an absolutely continuous invariant measure for local diffeomorphisms, only assuming the existence of hyperbolic times for Lebesgue almost all point of the manifold.
We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor.As a consequence under a mild assumption we prove exponential large deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.
Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $μ_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $Γ(t)=\int ϕdμ_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)=\sum_{k=0}^{\infty} \frac{v_t(f_t^k(c))}{Df_t^k(f_t(c))} \neq 0$ for all $t$, where $v_t(x)=\partial_s f_s(x)|_{s=t}$, then $Γ(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continu…
We present conditions on families of diffeomorphisms that guarantee statistical stability and SRB entropy continuity. They rely on the existence of horseshoe-like sets with infinitely many branches and variable return times. As an application we consider the family of Henon maps within the set of Benedicks-Carleson parameters.
We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y) = (E (x), C(y) + f(x) )$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$, $f$ is in $C^r(\mathbb{T}^u,\mathbb{R}^d)$ and $r \geq 2$. We prove that if $|(\det C)(\det E)| \|C^{-1}\|^{-2s}>1$ for some $s<r-(\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $…
For every $C^2$-small function $B$, we prove that the map $(x,y)\mapsto (x^2+a,0)+B(x,y,a)$ leaves invariant a physical, SRB probability measure, for a set of parameters $a$ of positive Lebesgue measure. When the perturbation $B$ is zero, this is the Jakobson Theorem; when the perturbation is a small constant times $(0,x)$, this is the celebrated Benedicks-Carleson Theorem. In particular, a new proof of the last theorem is given, based on devellopment of the combinatorial formalism of the Yoccoz puzzles. By adding new geometrical and combinatorial ingredients, and restructuring classic analyti…
We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated normalization, converges in distribution to the normal distribution.
We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >4, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1…
We study the entropy and Lyapunov exponents of invariant measures $μ$ for smooth surface diffeomorphisms $f$, as functions of $(f,μ)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion f…
Let $A$ be an expanding endomorphism on the torus ${\Bbb T}^d = {\Bbb R}^d /{\Bbb Z}^d$ with its smallest eigenvalue $λ>1$. Consider the ergodic system $({\Bbb T}^d, A, μ)$ where $μ$ is Haar measure. We prove that the correlation $ρ_{f, g}(n)$ of a pair of functions $f, g \in L^2(μ)$ is controlled by the modulus of $L^2$-continuity $Ω_{f, 2}(λ^{-n})$ and that the estimate is to some extent optimal. We also prove the central limit theorem for the stationary process $f(A^n x)$ defined by a function $f$ satisfying $Σ_n Ω_{f,2}(λ^{-n}) <\infty$. An application is given to the Ulam-von Neumann syst…
We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, which was recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion we count the number of orbits for the action of the subgroup of fixed points (resp. periodic) points on the set of infinite fixed (resp. periodic) points. Finally, we study the dynamics of infinite points: f…
We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous paper, but require proving Holder continuity of the infinitesimal conjugacy (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order if f_t is smooth enough and stays in the topological class of f_0 and if the observable smooth enough (a new result). In addition, this proof d…
The folding entropy is a quantity originally proposed by Ruelle in 1996 during the study of entropy production in the non-equilibrium statistical mechanics. As derived through a limiting process to the non-equilibrium steady state, the continuity of entropy production plays a key role in its physical interpretations. In this paper, we study the continuity of folding entropy for a general (non-invertible) differentiable dynamical system with degeneracy. By introducing a notion called degenerate rate, we prove that on any subset of measures with uniform degenerate rate, the folding entropy, and …
Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $μ$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,μ)$. Our class $\cal{U}$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $ω: (Σ, s, ν) \to (\mathbb{P}^k,f,μ)$. We obtain the invariance principle for an observable $ψ$ on $(\mathbb{P}^k,f,μ)$ by applying Philipp-Stout's theorem for $χ= ψ\circ ω$ on $(Σ, s, ν)$. The invariance principle implies…
Let $π:X\to Y$ be a factor map, where $(X,σ_X)$ and $(Y,σ_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous function on $X$ with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure $μ$ on $X$ that maximizes $μ(f)+a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $μ$ on $X$ that maximizes the weighted entropy $a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y…
