Related papers: Characterization of hyperbolic potentials

Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a H\"older continuous potential $\phi: J(f)\rightarrow \R$ and the pressure $P(f,\phi). In the case where $\sup_{J(f)}\phi<P(f,phi)$, the uniqueness and…

Consider a topologically exact $C^3$ interval map without non-flat critical points. Following the works we did in \cite{LiRiv12two}, we give two equivalent characterizations of hyperbolic H\"{o}lder continuous potential in terms of the…

We develop the specification and orbit-decomposition approach to equilibrium states for parabolic rational maps of the Riemann Sphere. Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably…

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\phi$ with he `bounded range' condition $\sup \phi - \inf \phi < \htop$, first used by…

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.…

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 $C^{\alpha}([0,1],\mathcal{P})$ the set 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,…

We consider a wide family of non-uniformly expanding maps and hyperbolic H\"older continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer…

We prove robustness and uniqueness of equilibrium states for a class of partially hyperbolic diffeomorphisms with dominated splittings and H\"older continuous potentials with not very large oscillation.

Equilibrium states are natural dynamical analogues of Gibbs states in thermodynamic formalism. This paper investigates their computability within the framework of Computable Analysis. We show that the unique equilibrium state for a…

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every H\"older continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems…

Let $\Lambda$ be a compact locally maximal invariant set of a $C^2$-diffeomorphism $f:M\to M$ on a smooth Riemannian manifold $M$. In this paper we study the topological pressure $P_{\rm top}(\phi)$ (with respect to the dynamical system…

Given an one-dimensional Lorenz-like expanding map we prove that the condition\linebreak $P_{top}(\phi,\partial \mathcal{P},\ell)<P_{top}(\phi,\ell)$ (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in [1] is satisfied…

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $\mu_\phi$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued H\"older continuous potential $\phi$.…

We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential $\varphi$ is computed by considering only those $(n,\epsilon)$-separated sets whose…

We establish the conditioned stochastic stability of equilibrium states for H\"older potentials on uniformly hyperbolic sets. While standard stochastic stability characterises measures on attractors, we analyse the statistics of transient…

For full shifts on finite alphabets, Coelho and Quas showed that the map that sends a H\"older continuous potential $\phi$ to its equilibrium state $\mu_\phi$ is $\overline{d}$-continuous. We extend this result to the setting of full shifts…

We study singular value potentials of H\"older continuous $GL_2(\mathbb{R})$-cocycles over hyperbolic systems whose canonical holonomies converge and are H\"older continuous. Such cocycles include locally constant…

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the…

We study the thermodynamic formalism of a $C^{\infty}$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a H\"older continuous potential with one additional condition, or the geometric t-potential…