Conditional Variational Principle Research Articles

A conditional variational principle for pressure of covers with respect to a partition

A conditional variational principle for pressure of covers with respect to a partition

A local conditional variational principle of pressures and local conditional equilibrium states

In this article, we introduce the notions of topological conditional pressures and two measure-theoretical conditional pressures of a finite open cover conditioned by a fixed finite measurable partition. We establish a local conditional variational principle and show that local conditional pressures determine local conditional measure-theoretical entropies. As applications, we study properties of both local and global conditional equilibrium states for continuous, real-valued functions.

Entropy via preimage structure

To measure the complexity of a “non-invertible” continuous map f on a compact metric space via the preimage structure, Hurley introduced the notion of pointwise topological preimage entropies hm(f) and hp(f) in 1995. A natural question was: Can one introduce the counterpart of hm(f) or hp(f) from the measure-theoretic point of view for an f-invariant measure μ and obtain properties analogous to that for the classical entropies? Cheng and Newhouse made the first step on this topic and introduced the notions of preimage entropies hpre(f) and hpre,μ(f) and investigated their properties in 2005. Recently, Wu and Zhu proposed a notion of pointwise metric preimage entropy hm,μ(f) and obtained some properties for f with uniform separation of preimages.In this paper, we give a complete answer to the above question for general continuous maps. The Shannon-McMillan-Breiman theorem for hm,μ(f) and the variational principle relating hm,μ(f) and hm(f) are obtained. It is shown that hm(f)=hpre(f) and the three types of measure-theoretic entropy-like invariants hpre,μ(f), hm,μ(f) and the folding entropy Fμ(f) (introduced by Ruelle) via the preimage structure coincide with each other. We also show that the preimage entropy variational principle can be realized as a variant of the conditional variational principle. These results answer several open questions asked by Cheng-Newhouse and Downarowicz. The upper semi-continuity of hm,μ(f) with respect to invariant measures with the same separation rate is obtained. Moreover, the preimage pressure and preimage equilibrium states are investigated.

Dimension spectra for flows: Future and past

We establish a conditional variational principle for the dimension spectrum obtained from almost additive families for a flow on a conformal locally maximal hyperbolic set, simultaneously into the future and into the past.

Conditional variational principles of conditional entropies for amenable group actions * *The work was supported by NSF of China (11671058, 12071049) and 2020CDCGJ019.

Let G be an infinite discrete countable amenable group acting continuously on two compact metrizable spaces X, Y. Assume that φ : (Y, G) → (X, G) is a factor map. Using finite open covers, the conditional topological entropy of φ is defined. The conditional measure-theoretic entropy of φ equals the conditional measure-theoretic entropy of Y to X. With the aid of tiling system of G, the conditional variational principle of φ is studied when (X, G) is an asymptotically h-expansive system. If X = Y and φ is the identity map, the conditional topological entropy of system (X, G) is defined. In the Cartesian square (X × X, G), we define the conditional measure-theoretic entropy of (X, G) to be the defect of the upper semi-continuity of the conditional measure-theoretic entropy of X × X to the first axis. Then the conditional variational principle of (X, G) is obtained.

Almost additive multifractal analysis for flows * *Partially supported by FCT/Portugal through UID/MAT/04459/2019. CH was supported by FCT/Portugal through the Grant PD/BD/135523/2018.

Building on the construction of equilibrium measures, we establish a conditional variational principle for the multifractal spectra of an almost additive family with respect to a continuous flow Φ such that the entropy map μ ↦ h μ (Φ) is upper-semicontinuous. We also show that the spectrum is continuous and that in the case of hyperbolic flows the corresponding irregular sets have full topological entropy. More generally, we consider the spectrum for the u-dimension and obtain corresponding results.

The conditional variational principle for maps with the pseudo-orbit tracing property

Let $(X,d,f)$ be a topological dynamical system, where $(X,d)$ is a compact metric space and $f:X \to X$ is a continuous map. We define $n$-ordered empirical measure of $x \in X$ by \begin{document}$\mathscr{E}_n(x) = \frac{1}{n}\sum\limits_{i = 0}^{n-1}δ_{f^ix},$ \end{document} where $δ_y$ is the Dirac mass at $y$. Denote by $V(x)$ the set of limit measures of the sequence of measures $\mathscr{E}_n(x)$. In this paper, we obtain conditional variational principles for the topological entropy of \begin{document}$\Delta_{sub}(I): = \left\{ {x \in X:V(x)\subset I} \right\},$ \end{document} and \begin{document}$\Delta_{cap}(I): = \left\{ {x \in X:V(x)\cap I≠\emptyset } \right\}.$ \end{document} in a dynamical system with the pseudo-orbit tracing property, where $I$ is a certain subset of $\mathscr M_{\rm inv}(X,f)$.

Topological entropy of irregular sets

For expansive continuous maps with the specification property, we compute the topological entropy of the irregular set for the Birkhoff averages of a continuous function. This is the set of points for which the Birkhoff averages do not converge. The entropy is expressed in terms of a conditional variational principle. We also consider the general case of irregular sets obtained from ratios of Birkhoff averages of continuous functions. Moreover, we obtain a conditional variational principle for the topological entropy of the family of subsets of the irregular set formed by the points such that the set of accumulation points of the ratio of Birkhoff averages is a given interval. As nontrivial applications, we obtain conditional variational principles for the topological entropy of the level sets of local entropies, pointwise dimensions and Lyapunov exponents both on repellers and hyperbolic sets.

On the topological pressure of the saturated set with non-uniform structure

We derive a conditional variational principle of the saturated set for systems with the non-uniform structure. Our result applies to a broad class of systems including $\beta$-shifts, $S$-gap shifts and their subshift factors.

Multifractal Analysis for Maps with the Gluing Orbit Property

In this paper, we obtain a conditional variational principle for the topological entropy of level sets of Birkhoff averages for maps with the gluing orbit property. Our result can be easily extended to flows.

Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems

This article is devoted to the studyof the irregular sets of Birkhoff averages in some nonuniformlyhyperbolic systems via Pesin theory.Particularly, we give a conditional variational principlefor the topological entropy of the irregular sets. Our result can be applied(i) to the diffeomorphisms on surfaces,(ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok.

Transience and multifractal analysis

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

Multifractal Analysis of the Asympyotically Additive Potentials

Multifractal analysis studies level sets of asymptotically defined quantities in dynamical systems. In this paper, we consider the u -dimension spectra on such level sets and establish a conditional variational principle for general asymptotically additive potentials by requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions.

Multifractal Analysis of Asymptotically Additive Sequences

For saturated maps, we effect a complete multifractal analysis of the dimension spectra obtained from asymptotically additive sequences of continuous functions. This includes, for example, the class of maps with the specification property. We consider also the more general cases of ratios of sequences and of multidimensional spectra in which a single sequence is replaced by a vector of sequences. In addition, we establish a conditional variational principle for the topological pressure of a continuous function on the level sets of an asymptotically additive sequence (again in the former general setting). Finally, we apply our results to the dimension spectra of an average conformal repeller. In particular, we obtain almost automatically a conditional variational principle for the Hausdorff dimension of the level sets obtained from an asymptotically additive sequence.

Topological pressure of simultaneous level sets

Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets.Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace.This technique allows us to extend a number of previous multifractal results from the C1+ε case to the C1 case. We consider ergodic ratios Snφ/Snψ where the function ψ need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.

Infinite non-conformal iterated function systems

We consider a class of iterated function systems consisting of a countable infinity of non-conformal contractions, extending both the self-affine limit sets of Lalley and Gatzouras as well as the infinite iterated function systems of Mauldin and Urbanski. Natural examples include the sets of points in the plane obtained by taking the binary expansion along the vertical and the continued fraction expansion along the horizontal and deleting certain pairs of digits. We prove that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures, which are given by a Ledrappier and Young type formula. In addition we consider the multifractal analysis of Birkhoff averages for countable families of potentials. We obtain a conditional variational principle for the level sets.

Multifractal analysis for Birkhoff averages on Lalley–Gatzouras repellers

We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Dimension spectra of almost additive sequences

We establish a conditional variational principle for the dimension spectra of almost additive sequences, considering simultaneously Birkhoff averages into the future and into the past. Our approach is based on the existence and uniqueness of equilibrium and Gibbs measures for the class of almost additive sequences, in the context of the nonadditive thermodynamic formalism. This formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function is replaced by the topological pressure of a sequence of functions.

On the variational principle for the topological pressure for certain non-compact sets

Let (X, d, T) be a dynamical system, where (X, d) is a compact metric space and T: X → X is a continuous map. We assume that the dynamical system satisfies g-almost product property and the uniform separation property. We compute the topological pressure of saturated sets under these two conditions. If the uniform separation property does not hold, we compute the topological pressure of the set of generic points. We give an application of these results to multifractal analysis and finally get a conditional variational principle.