In this paper, we investigate the computability of 𝒢-Bernoulli measures, with a particular focus on measures of maximal entropy (MMEs) on coded shift spaces. Coded shifts are natural generalizations of sofic shifts and are defined as the closure of all bi-infinite concatenations of words (generators) drawn from a countable generating set 𝒢. We begin by establishing a computability criterion for 𝒢-Bernoulli measures which are invariant measures given by assigning probability weights to the generators. We then apply this criterion to the setting in which the concatenation entropy exceeds the residual entropy, showing that in this case the unique measure of maximal entropy μ_ max on X is computable, provided the Vere–Jones parameter κ of 𝒢 is computable, based on having oracle access to the generators and the language of X. As a consequence, the unique MME is computable for several well-known classes of shift spaces, including S-gap shifts, multiple-gap shifts, and β-shifts. Moreover, the two ergodic MMEs of the Dyck shift are also computable. Finally, we examine the opposite situation, where the residual entropy exceeds the concatenation entropy and the MME is known to be non-unique in general. We show that even when μ_ max is unique and the parameter κ is computable, the measure μ_ max may still fail to be computable.

In this paper we present an introduction to the area of computability in dynamical systems. This is a fairly new field which has received quite some attention in recent years. One of the central questions in this area is if relevant dynamical objects can be algorithmically presented by a Turing machine. After providing an overview of the relevant objects we discuss recent results concerning the computability of the entropy for symbolic systems and the computability of Julia sets as well as their Brolin-Lyubich measures.

We study ergodic-theoretic properties of coded shift spaces. A coded shift space is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of Hölder continuous potentials based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). In this case we provide a simple explicit description of the measure of maximal entropy. We also obtain flexibility results for the entropy on the sequential and residual set. Finally, we prove a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov.

The statistical mechanics of molecular collisions in a macroscopic body is encoded by the parameter Thermodynamic entropy - a statistical measure of the number of molecular configurations that correspond to a given macrostate. Irreversibility in the flow of energy in macroscopic bodies is determined by random molecular interactions and expressed by the Second Law of Thermodynamics: In isolated systems, that is systems closed to the input of energy and matter, thermodynamic entropy increases. The evolutionary dynamics of a population of replicating organisms is encoded by the parameter Evolutionary entropy, a statistical measure of the number and diversity of metabolic cycles generated by the replication and viability of the individual organisms. Irreversibility in the transformation of energy in populations of organisms is determined by random mutation and natural selection. Changes in the organization of metabolic energy are described by the Directionality Principle of Evolution: Evolutionary entropy increases, when the energy source is limited and constant and decreases when the energy source is abundant and inconstant. This article shows that, when R -> 0, and N -> infinity (where R denotes the production rate of the external energy source; N the number of replicating organisms) evolutionary entropy, a measure of spatio-temporal organization, and thermodynamic entropy, a measure of positional disorder, coincide. We invoke this relation between thermodynamic and evolutionary entropy to show that the Directionality Principle for Evolution is a generalization of the Second Law of Thermodynamics. Published by Elsevier B.V.

We investigate the computability (in the sense of computable analysis) of the topological pressure P top ( ϕ ) on compact shift spaces X for continuous potentials ϕ : X → R . This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function ( X , ϕ ) ↦ P top ( X , ϕ | X ) is not computable for a large set of shift spaces X and potentials ϕ . In particular, the entropy map X ↦ h top ( X ) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

In this note we study the entropy spectrum of rotation classes for collections of finitely many continuous potentials φ1,…,φm:X→R with respect to the set of invariant measures of an underlying dynamical system f:X→X. We show for large classes of dynamical systems and potentials that these entropy spectra are maximal in the sense that every value between zero and the maximum is attained. We also provide criteria that imply the maximality of the ergodic entropy spectra. For m being large, our results can be interpreted as a complimentary result to the classical Riesz representation theorem in the dynamical context.

We study the regularity of the entropy spectrum of the Lyapunov exponents for hyperbolic maps on surfaces. It is well-known that the entropy spectrum is a concave upper semi-continuous function which is analytic on the interior of the set Lyapunov exponents. In this paper we construct a family of horseshoes with a discontinuous entropy spectrum at the boundary of the set of Lyapunov exponents.

Let ϕ:X→R be a continuous potential associated with a symbolic dynamical system T:X→X over a finite alphabet. Introducing a parameter β>0 (interpreted as the inverse temperature) we study the regularity of the pressure function β↦Ptop(βϕ) on an interval [α,∞) with α>0. We say that ϕ has a phase transition at β0 if the pressure function Ptop(βϕ) is not differentiable at β0. This is equivalent to the condition that the potential β0ϕ has two (ergodic) equilibrium states with distinct entropies. For any α>0 and any increasing sequence of real numbers (βn) contained in [α,∞), we construct a potential ϕ whose phase transitions in [α,∞) occur precisely at the βn's. In particular, we obtain a potential which has a countably infinite set of phase transitions.

Let $f:X\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $\mu\mapsto h_\mu(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $\phi:X\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $\Phi:X\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\Phi$ with a discontinuous localized entropy function.

The dynamics of molecular collisions in a macroscopic body are encoded by the parameter Thermodynamic entropy - a statistical measure of the number of molecular configurations that correspond to a given macrostate. Directionality in the flow of energy in macroscopic bodies is described by the Second Law of Thermodynamics: In isolated systems, that is systems closed to the input of energy and matter, thermodynamic entropy increases. The dynamics of the lower level interactions in populations of replicating organisms is encoded by the parameter Evolutionary entropy, a statistical measure which describes the number and diversity of metabolic cycles in a population of replicating organisms. Directionality in the transformation of energy in populations of organisms is described by the Fundamental Theorem of Evolution: In systems open to the input of energy and matter, Evolutionary entropy increases, when the energy source is scarce and diverse, and decreases when the energy source is abundant and singular. This article shows that when rho to 0, and N to infinity, where rho is the production rate of the external energy source, and N denote the number of replicating units, evolutionary entropy, an organized state of energy; and thermodynamic entropy, a randomized state of energy, coincide. Accordingly, the Fundamental Theorem of Evolution, is a generalization of the Second Law of Thermodynamics.

In this paper, we investigate the computability of thermodynamic invariants at zero temperature for one-dimensional subshifts of finite type. In particular, we prove that the residual entropy (i.e., the joint ground state entropy) is an upper semi-computable function on the space of continuous potentials, but it is not computable. Next, we consider locally constant potentials for which the zero-temperature measure is known to exist. We characterize the computability of the zero-temperature measure and its entropy for potentials that are constant on cylinders of a given length k. In particular, we show the existence of an open and dense set of locally constant potentials for which the zero-temperature measure can be computationally identified as an elementary periodic point measure. Finally, we show that our methods do not generalize to treat the case when k is not given

Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.

We provide a topological classification of locally constant functions over subshifts of finite type via their zero-temperature measures. Our approach is to analyze the relationship between the distribution of the zero-temperature measures and the boundary of higher dimensional generalized rotation sets. We also discuss the regularity of the localized entropy function on the boundary of the generalized rotation sets.

We consider a continuous dynamical system $f:X\to X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\Phi=(\phi_1,\cdots,\phi_m):X\to \bR^m$. We study the set of ground states $ GS(\alpha)$ of the potential $\alpha\cdot \Phi$ as a function of the direction vector $\alpha\in S^{m-1}$. %We also study the corresponding rotation vectors $\rv(GS(\alpha))$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\Phi$. In particular, for each $\alpha$ the set of rotation vectors of $ GS(\alpha)$ forms a non-empty, compact and connected subset of a face $F_\alpha(\Phi)$ of the rotation set associated with $\alpha$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_\alpha(\Phi)$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\in\bN$ examples with an exposed boundary point (i.e. $F_\alpha(\Phi)$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_\alpha(\Phi)$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\alpha)$ is a non-trivial line segment.

In this paper we study the dynamics of a family of diffeomorphisms in ^2 defined by F(x,y)=(g(x)+h(y),h(x)), where g(x) is a unimodal C^2-map which has the same dynamical properties as the logistic map P(x)=μ x(1-x), and h(x) is a C^2 map which is a small perturbation of a linear map. For certain maps of this form we show that there are exactly two periodic points, namely an attracting fixed point and a saddle fixed point and the boundary of the basin of attraction is the stable manifold of the saddle. The basin boundary also has the same regularity as F, in contrast to the frequently observed fractal nature of basin boundaries. To establish these results we describe the orbits under forward and backward iteration of every point in the plane.

Given a continuous dynamical system [Formula: see text] on a compact metric space [Formula: see text] and a continuous potential [Formula: see text], the generalized rotation set is the subset of [Formula: see text] consisting of all integrals of [Formula: see text] with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

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 statistical sums with respect to an m-dimensional potential \(\Phi \) are “close” to a given value \(w\in {\mathbb R}^m\). We then establish for several classes of systems and potentials \(\varphi \) and \(\Phi \) a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Hölder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, we construct an example with infinitely many ergodic localized equilibrium states. We also show that for systems with strong thermodynamic properties and w in the interior of the rotation set of \(\Phi \) there is at least one and at most finitely many localized equilibrium states.

For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (ϕ1, ..., ϕ m ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ℝ m . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ℝ m a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot Pt (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot Pt (Φ). Finally, we study the entropy function w ↦ H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ↦ H(w) is real-analytic in the interior of the rotation set.

Abstract We study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

For a rational map f on the Riemann sphere we study the entropy H(w) of points w in the barycenter set Omega(f). We show that this entropy is entirely determined by the growth rate of those repelling periodic orbits whose barycenters are close to w and that exhibit sufficient expansion. Assuming additionally that f is hyperbolic, we prove that H(w) is a real-analytic and strictly positive function on the interior of the barycenter set. We also consider the case of more general classes of potentials.