Unique equilibrium states for Viana maps for small potentials

We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that there is a unique equilibrium state in a sufficiently large external field.

We study the one parameter family of potential functions associated with the unstable Jacobian potential (or geometric potential) for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For q<1, it is known that there is a unique equilibrium state associated with , and it has full support. For q>1 it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value q = 1 and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure or measures supported on the singular set. In particular, when q = 1, there is a unique ergodic equilibrium state that gives positive measure to the regular set.

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a “periodic” external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T c and is non-unique forT<T c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.

It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed.

It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed.

This paper presents a comprehensive treatment of the existence, multiplicity, and qualitative behavior of equilibrium states for nonlinearly elastic strings under three different loading systems. It is shown that vertically loaded strings have unique convex equilibrium states and have concave equilibrium states that come in pairs. All equilibrium states are of one of these two forms. Similar, though more complicated, results hold for strings under normal pressure.

Given a finite irreducible set of real$d\times d$matrices$A_{1},\ldots ,A_{M}$and a real parameter$s&gt;0$, there exists a unique shift-invariant equilibrium state on$\{1,\ldots ,M\}^{\mathbb{N}}$associated to$(A_{1},\ldots ,A_{M},s)$. In this paper we characterize the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterize when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by$A_{1},\ldots ,A_{M}$, and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by Kusuoka are explored in an appendix.

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 Hölder continuous potential with one additional condition, or geometric $t$-potential $\unicode[STIX]{x1D711}_{t}$ with $t&lt;1$, the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$. We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.

By generalizing Ledrappier's criterion [Mesures d'èquilibre d'entropie complètement positive, in Systèmes dynamiques II – Varsovie, number 50 in Astérisque, Société mathématique de France, 1977, pp. 251–272] for the K-property of equilibrium states, we extend the criterion to subadditive potentials. In particular, supposing that the unique equilibrium state for a subadditive potential with quasi-multiplicativity and bounded distortion is totally ergodic, we show that it has the K-property. We apply this result to subadditive potentials arising from certain classes of matrix cocycles; for the norm potentials of irreducible locally constant cocycles and the singular value potentials of typical cocycles, we show that their unique equilibrium states have the K-property. This partly generalizes the work of Morris [Ergodic properties of matrix equilibrium states, Ergodic Theory Dyn. Syst. 38(6), 2018, pp. 2295-2320] on irreducible locally constant cocycles and their subadditive equilibrium states.

As with probability theory, uncertainty theory has been developed, in recent years, to portray indeterminacy phenomena in various application scenarios. We are concerned, in this paper, with the convergence property of state trajectories to equilibrium states (or fixed points) of time delayed uncertain cellular neural networks driven by the Liu process. By applying the classical Banach’s fixed-point theorem, we prove, under certain conditions, that the delayed uncertain cellular neural networks, concerned in this paper, have unique equilibrium states (or fixed points). By carefully designing a certain Lyapunov–Krasovskii functional, we provide a convergence criterion, for state trajectories of our concerned uncertain cellular neural networks, based on our developed Lyapunov–Krasovskii functional. We demonstrate under our proposed convergence criterion that the existing equilibrium states (or fixed points) are exponentially stable almost surely, or equivalently that state trajectories converge exponentially to equilibrium states (or fixed points) almost surely. We also provide an example to illustrate graphically and numerically that our theoretical results are all valid. There seem to be rare results concerning the stability of equilibrium states (or fixed points) of neural networks driven by uncertain processes, and our study in this paper would provide some new research clues in this direction. The conservatism of the main criterion obtained in this paper is reduced by introducing quite general positive definite matrices in our designed Lyapunov–Krasovskii functional.

In this paper equilibrium states on shift spaces are considered. A uniqueness theorem for equilibrium states is proved. Then we study a particular class of continuous functions. We characterize the functions of this class which satisfy Ruelle’s Perron-Frobenius condition, those which admit a measure determined by a homogeneity condition, and those which have unique equilibrium state. In particular, we get examples for the nonuniqueness of the equilibrium state.

It is well known that for expansive maps and continuous potential functions, the specification property (for the map) and the Bowen property (for the potential) together imply the existence of a unique equilibrium state. We consider symbolic spaces that may not have specification and potentials that may not have the Bowen property, and give conditions under which uniqueness of the equilibrium state can still be deduced. Our approach is to ask that the collection of cylinders which are obstructions to the specification property or the Bowen property is small in an appropriate quantitative sense. This allows us to construct an ergodic equilibrium state with a weak Gibbs property, which we then use to prove uniqueness. We do not use inducing schemes or the Perron–Frobenius operator, and we strengthen some previous results obtained using these approaches. In particular, we consider β-shifts and show that the class of potential functions with unique equilibrium states strictly contains the set of potentials with the Bowen property. We give applications to piecewise monotonic interval maps, including the family of geometric potentials, for example, which have both indifferent fixed points and a non-Markov structure.

Author(s): Li, Zhiqiang | Advisor(s): Bonk, Mario | Abstract: Thurston maps are topological generalizations of postcritically-finite rational maps. More precisely, a Thurston map $f\colon S^2\rightarrow S^2$ is a (non-homeomorphic) branched covering map on a topological 2-sphere $S^2$ whose critical points are all preperiodic. This thesis provides a comprehensive study of the measure of maximal entropy, as well as a more general class of invariant measures, called equilibrium states, for expanding Thurston maps. In particular, given an expanding Thurston map $f\colon S^2 \rightarrow S^2$, we present a large class of equidistribution results for iterated preimages and (pre)periodic points with respect to the unique measure of maximal entropy by first establishing a formula for the number of fixed points. The formula states that $f$ has exactly $1+\operatorname{deg} f$ fixed points, counted with appropriate weights, where $\operatorname{deg} f$ denotes the topological degree of the map $f$. We then use the thermodynamical formalism to show that there exists a unique equilibrium state $\mu_\phi$ for $f$ together with a real-valued H quot;{o}lder continuous potential $\phi$. 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 $(S^2,\mu_\phi)$ is exact, and in particular, mixing and ergodic. Moreover, we establish a version of equidistribution of a random backward orbit with respect to the equilibrium state. After proving that $f$ is asymptotically $h$-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is $h$-expansive, we obtain certain large deviation principles for iterated preimages and periodic points under the additional assumption that $f$ has no periodic critical points. This enables us to obtain general equidistribution results for iterated preimages and periodic points with respect to the equilibrium states under the same assumption on $f$. We also get the existence of equilibrium states for such $f$ and any continuous real-valued potential.

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 computable, open, topologically exact, distance-expanding map $T\colon X\rightarrow X$ and a computable Hölder continuous potential $\varphi \colon X\rightarrow \mathbb{R}$ is always computable. As an application, we establish the computability of equilibrium states for computable hyperbolic rational maps and their respective geometric potentials. Moreover, we develop a constructive method to exhibit the non-uniqueness of equilibrium states for some dynamical systems. We also present some computable dynamical systems whose equilibrium states are all non-computable.

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.