We develop a quenched thermodynamic formalism for open random dynamical systems generated by finitely branched, piecewise-monotone mappings of the interval. The openness refers to the presence of holes in the interval, which terminate trajectories once they enter; the holes may also be random. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$. For each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$, and a hole $H_ω\subset I$; the map $T_ω$, the random potential $φ_ω$, and the hole $H_ω$ generate the corresponding open transfer operator $\mathcal{L}_ω$. For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani--Maume-Deschamps, we prove there exists a unique random probability measure $ν_ω$ supported on the survivor set ${X}_{ω,\infty}$ satisfying $ν_{σ(ω)}(\mathcal{L}_ωf)=λ_ων_ω(f)$. We also prove the existence of a unique random family of functions $q_ω$ that satisfy $\mathcal{L}_ωq_ω=λ_ωq_{σ(ω)}$. These yield an ergodic random invariant measure $μ=νq$ supported on the global survivor set, while $q$ combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) $η$ supported on $I$. We prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $μ$. Finally, the escape rates of the random closed conformal measure and the RACCIM $η$ coincide, and are given in terms of the expected pressure, as is the Hausdorff dimension of the surviving set $X_{ω,\infty}$. We provide examples of our general theory, including random $β$-transformations and random Lasota-Yorke maps.
We study ergodic properties of certain piecewise smooth two-dimensional systems by constructing countable Markov partitions. Using thermodynamic formalism we prove exponential decay of correleations.
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.
We introduce a higher-dimensional version of the nonlinear thermodynamic formalism introduced by Buzzi and Leplaideur, in which a potential is replaced by a family of potentials. In particular, we establish a corresponding variational principle and we discuss the existence, characterization, and number of equilibrium measures for this higher-dimensional version.
This paper is devoted to study how do thermodynamic formalism quantities varies for time changes of suspension flows defined over countable Markov shifts. We prove that in general no quantity is preserved. We also make a topological description of the space of suspension flows according to certain thermodynamic quantities. For example, we show that the set of suspension flows defined over the full shift on a countable alphabet having finite entropy is open. Of independent interest might be a set of analytic tools we use to construct examples with prescribed thermodynamic behaviour.
We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion we construct new Markov rectangles such that their crossections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain Hölder functions. The results are based on the methods developed by Sarig.
We study the expansion properties of the contracting Lorenz flow introduced by Rovella via thermodynamic formalism. Specifically, we prove the existence of an equilibrium state for the natural potential $\hatϕ_t(x,y, z):=-t\log J_{(x, y, z)}^{cu}$ for the contracting Lorenz flow and for $t$ in an interval containing $[0,1]$. We also analyse the Lyapunov spectrum of the flow in terms of the pressure.
We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T\colon \RR\to \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formal…
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.
Working with well chosen Riemannian metrics and employing Nevanlinna's theory, we make the thermodynamical formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine--root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynam…
We construct measures of maximal $u$-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has a finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.
We introduce a definition of pressure for almost-additive sequences of continuous functions defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs and equilibrium measures. Applications are given to the study of maximal Lypaunov exponents of product of matrices.
Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.
We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and positive Radon eigenmeasures in forms of Martin kernels. These eigenmeasures can be characterized in terms of the direction of escape to infinity of their orbits, when viewed inside a suitable Martin-like compactification of the underline shift space. We relate these results to first-order phase transitions in one-dimensional lattice gas models with infinite set of states. This work complements ea…
In this note we study the thermodynamic formalism for the positive geodesic flow on the modular surface. We define the pressure and prove the variational principle. We also establish conditions for the the pressure to be real analytic and for the potentials to have unique equilibrium states. The results in this paper were largely superceded by "Phase transitions for suspension flows" by Iommi and Jordan arXiv:1202.0849
We study a one-parameter family of countably piecewise linear interval maps, which, although Markov, fail the `large image property'. This leads to conservative as well as dissipative behaviour for different maps in the family with respect to Lebesgue. We investigate the transition between these two types, and study the associated thermodynamic formalism, describing in detail the second order phase transitions (i.e. the pressure function is $C^1$ but not $C^2$ at the phase transition) that occur in transition to dissipativity. We also study the various natural definitions of pressure which ari…
In this paper we define distance expanding random dynamical systems. We develop the appropriate thermodynamic formalism of such systems. We obtain in particular the existence and uniqueness of invariant Gibbs states, the appropriate pressure function and exponentially fast convergence of iterates of Perron-Frobenius operators resulting, in particular, in an exponential decay of correlations. We also obtain the formula for the derivative of the expected value of the pressure function. Next, we define and investigate in detail conformal random expanding repellers. Applying the developed machiner…
For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any Hölder continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.
We study singular value potentials of Hölder continuous $GL_2(\mathbb{R})$-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant $GL_2(\mathbb{R})$-cocycles as well as fiber-bunched $GL_2(\mathbb{R})$-cocycles. We show that singular value potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose singular value potentials have more than one equilibrium states.
For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentia…
We survey some results on non-uniform hyperbolicity, geometric pressure and equilibrium states in one-dimensional real and complex dynamics. We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere. We discuss fluctuations of iterated sums of the potential $-t\log |f'|$ and of radial growth of derivative of univalent functions on the unit disc and the boundaries of range domains preserved by a holomorphic map $f$ repelling towards the domains.
In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of th…
This text provides an overview of the (geometric) thermodynamic formalism for transcendental meromorphic and entire functions with particular emphasis on geometric/fractal aspects such as Bowen's Formula expressing the hyperbolic dimension as a unique zero of a pressure function and the behavior of the latter when the transcendental functions vary in an analytic family.
This paper introduces an intrinsic theory of Thermodynamic Formalism for Iterated Functions Systems with general positive continuous weights (IFSw).We study the spectral properties of the Transfer and Markov operators and one of our first results is the proof of the existence of at least one eigenprobability for the Markov operator associated to a positive eigenvalue. Sufficient conditions are provided for this eingenvalue to be the spectral radius of the transfer operator and we also prove in this general setting that positive eigenfunctions of the transfer operator are always associated to i…
We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all s…
Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family $f_λ$ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential $ϕ_t = -t\log|f'_λ|$, there is a unique phase transition at some $t_1 \le 1$, and the pressure $P(ϕ_t)$ is analytic (with unique equilibrium state) elsewhere. The pressure is majorised…
This paper studies properties of a Renormalization Operator for potentials in symbolic dynamics. These operators first appeared in \cite{BLL} and the link with substitutions was done in \cite{BL1}. Their fixed points are natural candidates to have pathologic behavior such as phase transitions. If $R$ is such an operator, we study the convergence of $R^{n}(φ)$ to the non-nul fixed point. We define the family of marked substitutions, which contains the Thue-Morse substitution, and show that the associated renormalization operators on potentials admits a unique non-nul continuous fixed point. The…
In this paper we study the ergodic theory and thermodynamic formalism of the geodesic flow on non-compact pinched negatively curved manifolds. We consider two notions of entropy at infinity, the topological and the measure theoretic entropy at infinity. We prove that both notions coincide and we relate this quantity with the upper semicontinuity of the entropy map. This relationship was already studied in [RV] for geometrically finite manifolds; in this paper we extend such results to arbitrary pinched negatively curved manifolds. We obtain several applications to the existence and stability o…
We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential $φ$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a unique random conformal measure $ν_φ$ and unique random equilibrium state $μ_φ$. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for $μ_φ$. Our random driving is generated by an invertible, ergodic, measure-preserving transform…
We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator $\CR$ acting on the space of potentials are such that the pressure function $γ\mapsto \CP(-γV)$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attra…
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）
We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic $\mathbb{Z}$-extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the whole dimension spectrum with respect to $α$-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely …
We study $k$-bonacci substitutions. For each we define a renormalization operator associated to it and examine its iterates over potentials in a certain class. We also study the pressure function associated to potentials in this class and prove the existence of a freezing phase transition which is realized by the only ergodic measure on the subshift associated to the substitution.
We provide the full theory of thermodynamic formalism for a very general collection of entire functions in class $\mathcal B$. This class overlaps with the collection of all entire functions for which thermodynamic formalism has been so far established and contains many new functions. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quas…
We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {\it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1051-1082 (2004). We show only the uniqueness of the β-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $μ$) and its relation with the eigen-probability $ν_β$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for …
We introduce a non-commutative generalization of the notion of (approximately proper) equivalence relation and propose the construction of a "quotient space". We then consider certain one-parameter groups of automorphisms of the resulting C*-algebra and prove the existence of KMS states at every temperature. In a model originating from Thermodynamics we prove that these states are unique as well. We also show a relationship between maximizing measures (the analogue of the Aubry-Mather measures for expanding maps) and ground states. In the last section we explore an interesting example of phase…
This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional.
Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $Φ$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $Φ$. We setup the weighted variational principle. In the case that $X, Y$ are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional wei…
The aim of this study is tree-fold. First, we investigate the thermodynamics of the Ising models with respect to 2-multiple Hamiltonians. This extends the previous results of [Chazotte and Redig, Electron. J. Probably., 2014] to $\mathbb{N}^d$. Second, we establish the large deviation principle (LDP) of the average $\frac{1}{N} S_N^G$, where $S_N^G$ is a 2-multiple sum along a semigroup generated by k numbers which are k co-primes. This extends the previous results [Ban et al. Indag. Math., 2021] to a board class of the long-range interactions. Finally, the results described above are generali…
Thermodynamic formalism for hyperbolic random dynamical systems
Université de Toulon, Aix Marseille Université（土伦大学，艾克斯-马赛大学）；Centro de Matemática da Universidade do Porto（波尔图大学数学中心）；Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）；Aix Marseille Univ（艾克斯-马赛大学）；Great Bay University（大湾区大学）
In this paper, we study the thermodynamic formalism of interval maps $f$ with sufficient regularity, for a sub class $\mathcal{U}$ composed of upper semi-continuous potentials which includes both Hölder and geometric potentials. We show that for a given $u\in \mathcal{U}$ and negative values of $t$, the pressure function $P(f,-tu)$ can be calculated in terms of the corresponding hidden pressure function $\widetilde{P}(f,-tu)$. Determination of the values $t\in(-\infty,0)$ at which $P(f,-tu)\neq \widetilde{P}(f,-tu)$ is also characterized explicitly. When restricting to the Hölder continuous po…
We introduce a dissipative version of the Zhang's model of Self-Organized Criticality, where a parameter allows to tune the local energy dissipation. We analyze the main dynamical features of the model and relate in particular the Lyapunov spectrum with the transport properties in the stationary regime. We develop a thermodynamic formalism where we define formal Gibbs measure, partition function and pressure characterizing the avalanche distributions. We discuss the infinite size limit in this setting. We show in particular that a Lee-Yang phenomenon occurs in this model, for the only conserva…
