We introduce a class of continuous maps f of a compact metric space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions ϕon I which admit unique equilibrium measures μ_ϕminimizing the free energy for a certain class of measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the Central Limit Theorem. Our results apply in particular to some one-dimensional unimodal and multimodal maps as well as to multidimensional nonuniformly hyperbolic maps admitting Young's tower. Examples of potential functions to which our theory applies include ϕ_t=-t\log|df| with t\in(t_0, t_1) for some t_0<1<t_1. In the particular case of S-unimodal maps we show that one can choose t_0<0 and that the class of measures under consideration comprises all invariant Borel probability measures. Thus our results establish existence and uniqueness of both the measure of maximal entropy (by a different method than Hofbauer) and the absolutely continuous invariant measure extending results by Bruin and Keller for the parameters under consideration.
We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity.
This survey article concerns inducing schemes in the context of interval maps. We explain how the study of these induced systems allows for the fine description of, not only, the thermodynamic formalism for certain multimodal maps, but also of its multifractal structure.
For a smooth map f of a compact interval I admitting an inducing scheme we establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions $ϕ$ on I which admit a unique equilibrium measure $μ_ϕ$. Our results apply to unimodal maps corresponding to a positive Lebesgue measure set of parameters in a one-parameter transverse family.
For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure preserving) non-Markov intervals maps with an indifferent fixed point.
In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [PS05, PS06]. We show that under some natural assumptions on the inducing schemes - which hold for many known examples - any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [Buz99], his results on the liftability of measures of large entropy, and a generalization of…
We prove that return time statistics of a dynamical system do not change if one passes to an induced (i.e. first return) map. We apply this to show exponential return time statistics in i) smooth interval maps with nowhere-dense critical orbits and ii) certain interval maps with neutral fixed points. The method also applies to iii) certain quadratic maps of the complex plane.
We consider continuous maps $f:X\to X$ on compact metric spaces admitting inducing schemes of hyperbolic type introduced in [15] as well as the induced maps $\tilde{f}:\tilde{X}\to\tilde{X}$ and the associated tower maps $\hat{f}:\hat{X} \to \hat {X}$. For a certain class of potential functions $φ$ on $X$, which includes all Hölder continuous functions, we establish thermodynamic formalism for each of the above three systems and we describe some relations between the corresponding equilibrium measures. Furthermore we study ergodic properties of these equilibrium measures including the Bernoull…
We consider the thermodynamic formalism of a complex rational map $f$ of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter $t$ we study the (non-)existence of equilibrium states of $f$ for the potential $-t \ln |f'|$, and the analytic dependence on $t$ of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism of a rational map that is "expanding away from critical points" and that has arbitrarily small "nice sets" with some additional properties. Our results apply in particular…
We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual poin…
Given a multimodal interval map $f:I \to I$ and a Hölder potential $ϕ:I \to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $ϕ$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions.
We obtain entropy formulas for SRB measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to sev…
We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a smooth codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior.
We introduce a class of continuous maps $f$ of a compact topological space $X$ admitting inducing schemes of hyperbolic type and describe the associated tower constructions. We then establish a thermodynamic formalism, i.e., we describe a class of real-valued potential functions $φ$ on $X$ such that $f$ possess a unique equilibrium measure $μ_φ$, associated to each $φ$, which minimizes the free energy among the measures that are liftable to the tower. We also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem. We then study th…
We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.
We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing.
We consider families of multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential $ϕ_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these equilibrium states vary continuously in the weak$^*$ topology within such families. Moreover, in the case $t=1$, when the equilibrium states are absolutely continuous with respect to Lebesgue, we show that the densities vary continuously within these families.
Let $f:I \to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $ϕ_t:x\mapsto -t\log|Df(x)|$ for $t$ close to 1, and also that the pressure function $t \mapsto P(ϕ_t)$ is analytic on an appropriate interval near $t = 1$.
We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the Thermodynamical Formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.
We study the thermodynamic formalism of sufficiently regular interval maps for Holder continuous potentials. We show that for a hyperbolic potential there is a unique equilibrium state, and that this measure is exponentially mixing. Moreover, we show the absence of phase transitions: The pressure function is real analytic at such a potential.
We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a contracting foliation and inducing a piecewise expanding quotient map, with infinite derivative (like the first return maps of Lorenz like flows). We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error, with respect to the Wasserstein distance. We apply this to the rigorous computation of the dimension of the measure. We present a rigorous implementation of the algorithms using interval arithmetics, and the result of the computation …
We show a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a large class of maps with indifferent fixed points. It turns out that the $L^1$ dependence of the a.c.i.m. on small suitable deterministic changes for these kind of maps is Hölder, with an exponent which is explicitly estimated.
We show that, for a robust ($C^2$-open) class of random non-uniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu-Zhao. The proof of the main result uses an extension of techniques in recent works by Alves-Araújo, Alves-Bonatti-Viana and Oliveira.
We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen's formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite …
We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact manifolds and Holder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the …
In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on $\mathbb{T}^4$, with 2-dimensional center foliation. To do so we propose to study the disintegration of measures along 1-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context.
We study the existence, uniqueness and rate of decay of correlation of equilibrium measures associated to robust classes of non-uniformly expanding local diffeomorphisms and Hölder continuous potentials. The approach used in this paper is the spectral analysis of the Ruelle-Perron-Frobenius transfer operator. More precisely, we combine the expanding features of the eigenmeasures of the transfer operator with a Lasota-Yorke type inequality to prove the existence of an unique equilibrium measure with fast decay of correlations.
This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are …
We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (statistical stability). Counterexamples to statistical stability in the absence of strong hypotheses are provided.
For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of …
There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, "hyperbolic". We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Holder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Holder continuous functions. Another consequence is that every Holder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.
Thermodynamic formalism for discontinuous maps and statistical properties of their equilibrium states
Escuela de Matemática y Estadística, Universidad Pedagógica y Tecnológica de Colombia（哥伦比亚教育技术大学数学与统计学院）；Universidade Federal de Alagoas, Instituto de Matemática - UFAL（阿拉戈斯联邦大学数学研究所）
In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $μ_ϕ$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued Hölder continuous potential $ϕ$. Here the sphere $S^2$ is equipped with a natural metric induced by $f$, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation $f$ of the probability space $(S^2,μ_ϕ)$ is exact, and in particular, mixing and ergodic. Moreover, we establish ve…
We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $β\in [0, 1]$ and $α\in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $β$ of order $α$ is dependent on the structure of the $ω$-limit set of $β$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function cla…
An $S$-unimodal map $f$ with flat critical point satisfying the Misiurewicz condition displays a freezing phase transition in positive spectrum. We analyze statistical properties of the equilibrium state $μ_t$ for the potential $-t\log|Df|$, as well as how the phase transition slows down the rate of decay of correlations. We show that $μ_t$ has exponential decay of correlations for all inverse temperature $t$ contained in the positive entropy phase $(t^-,t^+)$. If the critical point is not too flat, then the freezing point $t^+$ is equal to $1$, and the absolutely continuous invariant probabil…
Let $L:[0,1]\setminus\{d\}\rightarrow [0,1]$ be a one-dimensional Lorenz like expanding map ($d$ is the point of discontinuity), $\mathcal{P}=\{ (0,d),(d,1) \}$ be a partition of $[0,1]$ and $C^α([0,1],\mathcal{P})$ the set of piecewise Hölder-continuous potential of [0,1] with the usual $\mathcal{C}^0$ topology. In this context, we prove, improving a result of \cite{BS03}, that piecewise Hölder-continuous potential $ϕ$ satisfying \linebreak $\max\left\{ \limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}ϕ)(0),\limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}ϕ)(1)\right\}<P_{\text{top}}(ϕ,T)$ suppor…
We develop a thermodynamic formalism for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as $t\to+\infty$ minimizes the unstable Lyapunov exponent. …
For any C1 expanding map f of the circle we study the equilibrium states for the potential -log |f'|. We formulate a C1 generalization of Pesin's Entropy Formula that holds for all the SRB measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C1-generic case Pesin's Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non generic case for which no SRB measure exists.
Effective shadowing of transition maps for a Hamiltonian saddle-center-center equilibrium
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）；Laboratory of Geometry and Dynamical Systems and IMTech, UPC（UPC几何与动力系统实验室和IMTech）；Department of Mathematical Sciences, Yeshiva University（叶史亚大学数学科学系）
We prove an equidistribution result for $C^{\infty}$ maps with respect to equilibrium states. We apply the result to the time-one map of the geodesic flow of a closed smooth Riemannian manifold.
We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψand a unique $\exp(¶(ψ)-ψ)$-conformal measure, where P(ψ) denotes the topological pressure of ψ.
We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle-Perron-Frobenius operator acting on the space of Holder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operat…
Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a Hölder continuous potential $ϕ: J(f)\rightarrow \R$ and the pressure $P(f,ϕ). In the case where $\sup_{J(f)}ϕ<P(f,phi)$, the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials $ϕ$ for which this property is satisfied for some iterate of $f$, in terms of the expanding properties of the corresponding equilibrium states. A direct consequence of this result is that for a nonuniformly hyperbolic rational map every …
Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks & Carleson to build induced Markov maps of…
Assume that $(X,f)$ is a dynamical system and $ϕ:X \to [-\infty, \infty)$ is a potential such that the $f$-invariant measure $μ_ϕ$ equivalent to $ϕ$-conformal measure is infinite, but that there is an inducing scheme $F = f^τ$ with a finite measure $μ_{\barϕ}$ and polynomial tails $μ_{\barϕ}(τ\geq n) = O(n^{-β})$, $β\in (0,1)$. We give conditions under which the pressure of $f$ for a perturbed potential $ϕ+sψ$ relates to the pressure of the induced system as $P(ϕ+sψ) = (C P(\overline{ϕ+sψ}))^{1/β} (1+o(1))$, together with estimates for the $o(1)$-error term. This extends results from Sarig to …
We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential…
We consider the case of hyperbolic basic sets $Λ$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $μ_ϕ$ associated to a class of Hölder potentials $ϕ$ on $Λ$, and find the measures $μ_ϕ$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $δ_{μ_ϕ}$ of $μ_ϕ$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $μ_ϕ$ in the case when the preimage counting function is constant on $Λ$. For terminal/minimal saddle sets we prove that an invariant…
