Sub-additive Potentials Research Articles

Variational principles for sub-additive unstable topological pressures on saturated subsets

In this paper, under the assumption of unstable almost product property for C 1-smooth partially hyperbolic diffeomorphisms, we establish variational principles for their unstable Bowen topological pressures and unstable packing topological pressures on the typical saturated subsets G K , where K is a given non-empty compact connected set of invariant measures. Actually, we show that these two dimension-like topological quantities coincide with the infimum and supremum respectively of the summation of unstable metric entropy and Lyapunov exponent, where the infimum and supremum are taken over all measures inside K. Besides, we also show that G K has full unstable topological capacity pressure for reasonable sub-additive potentials.

Double variational principle for mean dimensions with sub-additive potentials

In this paper, we introduce mean dimension quantities with sub-additive potentials. We define the mean dimension and mean metric dimension with sub-additive potentials and establish a double variational principle for sub-additive potentials.

Variational principle for weighted topological pressure with subadditive potentials

Variational principle for weighted topological pressure with subadditive potentials

Genericity of historic behavior for maps and flows

We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Mañé and Shub examples of robustly transitive non-hyperbolic diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications.

On the joint continuity of topological pressures for sub-additive singular-valued potentials

Given a non-conformal repeller [Formula: see text] of a [Formula: see text] map [Formula: see text], we give a variational principle of a dimensional upper bound of the non-conformal repellers. If [Formula: see text] is [Formula: see text] then we prove the joint continuity of topological pressure for sub-additive singular-valued potentials on maps and parameters.

The K-property for subadditive equilibrium states

ABSTRACT 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.

Unstable metric pressure of partially hyperbolic diffeomorphisms with sub-additive potentials

In this paper, we define and study unstable measure theoretic pressure for C 1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. We show that this measure theoretic pressure for any ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure. On the other hand, we also give other definitions of unstable metric pressure, in terms of Bowen’s picture and the capacity picture. We show that all definitions of unstable metric pressure, including the one defined at the beginning, actually coincide for any ergodic measure.

Zero temperature limits of equilibrium states for subadditive potentials and approximation of maximal Lyapunov exponent

In this paper we study ergodic optimization problems for subadditive sequences of functions on a topological dynamical system. We prove that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states is a maximizing measure. We show that the Lyapunov exponent and entropy of equilibrium states converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures. In the particular case of matrix cocycles we prove that the maximal Lyapunov exponent can be approximated by Lyapunov exponents of periodic trajectories under certain assumptions.

Relative Tail Pressure with Subadditive Potentials

The relative tail pressure with subadditive potentials is introduced via open random covers for continuous bundle random transformations. A variational principle is established and the defined pressure turns out to be invariant under the principal extension.

Equilibrium states of generalised singular value potentials and applications to affine iterated function systems

We completely describe the equilibrium states of a class of potentials over the full shift which includes Falconer's singular value function for affine iterated function systems with invertible affinities. We show that the number of distinct ergodic equilibrium states of such a potential is bounded by a number depending only on the dimension, answering a question of A. Kaenmaki. We prove that all such equilibrium states are fully supported and satisfy a Gibbs inequality with respect to a suitable subadditive potential. We apply these results to demonstrate that the affinity dimension of an iterated function system with invertible affinities is always strictly reduced when any one of the maps is removed, resolving a folklore open problem in the dimension theory of self-affine fractals. We deduce a natural criterion under which the Hausdorff dimension of the attractor has the same strict reduction property.

Sub-additive topological and measure-theoretic tail pressures

In this paper, we give some definitions of the topological tail pressures for sub-additive potentials and prove that they are equivalent if the potentials are continuous. Under some assumptions, we get a variational principle which exhibits the relationship between topological tail pressure and measure-theoretic tail entropy. Finally, we define a new measure-theoretic tail pressure for sub-additive potentials and some interesting properties of it are obtained.

Aubry set for asymptotically sub-additive potentials

Given a topological dynamical systems [Formula: see text], consider a sequence of continuous potentials [Formula: see text] that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set [Formula: see text] of [Formula: see text]-invariant probabilities that attain the following maximum value [Formula: see text] For this purpose, we extend the notion of Aubry set, denoted by [Formula: see text]. Our central result provides sufficient conditions for the Aubry set to be a maximizing set, i.e. [Formula: see text] belongs to [Formula: see text] if, and only if, its support lies on [Formula: see text]. Furthermore, we apply this result to the study of the joint spectral radius in order to show the existence of periodic matrix configurations approaching this value.

Variational principle for topological tail pressures with sub-additive upper semi-continuous potentials

ABSTRACTThe topological tail pressure and the conditional pressure are defined for sub-additive upper semi-continuous potentials via separated subsets and refining sequences of partitions in topological dynamical systems with finite topological entropy respectively. By proving the equivalence of the topological tail pressure and the conditional pressure with respect to the refining sequence of essential partitions, we present a variational principle for the topological tail pressure without any additional assumptions. The properties of the topological tail pressure including the power rule, the product rule and the natural extension are also discussed.

Sub-additive pressure on a borel set

The goal of this paper is to investigate topological conditional pressure of a continuous transformation as defined for sub-additive potentials. This study presents a variational inequality for sub-additive topological conditional pressure on a closed subset, which is the other form of the variational principle for the sub-additive topological pressure presented by Cao, Feng, and Huang in [9]. Moreover, under some additional assumptions, this result can be generalized to the non-compact case.

Modeling potential as fiber entropy and pressure as entropy

AbstractWe first prove that topological fiber entropy potential in a relatively symbolic extension (topological joining with a subshift) of a topological dynamical system (which is a non-negative, non-decreasing, upper semicontinuous and subadditive potential$\mathfrak{H}$) yields topological pressure equal to the topological entropy of the extended system. The terms occurring on the other side of the variational principle for pressure are equal to the extension entropies of the invariant measures. Thus the variational principle for pressure reduces to the usual variational principle (for entropy) applied to the extended system. Next we prove our main theorem saying that every non-negative, upper semicontinuous and subadditive potential$\mathfrak{F}$(we drop the monotonicity assumption) is ‘nearly equal’ to the fiber entropy potentialHin some relatively symbolic extension of the system, in the sense that all terms occurring in the variational principle for pressure are the same for both potentials. This gives a new interpretation of all such potentials$\mathfrak{F}$as a kind of additional information function enhancing the natural information arising from the dynamical system, and provides a new proof of the variational principle for pressure. At the end of the paper we provide examples showing that both assumptions, continuity and additivity, under which so-called lower pressure (defined with the help of spanning sets) equals the pressure, are essential, already in the class of non-negative, upper semicontinuous, subadditive potentials.

COSET PRESSURE WITH SUB-ADDITIVE POTENTIALS

The goal of this paper is to define the coset topological pressure for sub-additive potentials via separated sets on a compact metric group. Analogues of basic properties for topological pressure hold. This study also reveals a variational principle for the coset topological pressure. The process of the proof is quite similar to that of Cao, Feng and Huang's approximations, but the analysis needs more techniques of ergodic theory and topological dynamics.

Weak specification properties and large deviations for non-additive potentials

AbstractWe obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given.

Sub-Additive and Asymptotically Sub-Additive Topological Pressure for $$\mathbb{Z }^d$$ -Actions

We introduce the topological pressure for any sub-additive and asymptotically sub-additive potentials of $$\mathbb{Z }^d$$ -actions, and establish the variational principle for them.

Variational principles for relative local pressure with subadditive potentials

We prove two relative local variational principles of topological pressure functions \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U},y)$\end{document}P(T,F,U,y) and \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U}|Y)$\end{document}P(T,F,U|Y) for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover \documentclass[12pt]{minimal}\begin{document}$\mathcal {U}$\end{document}U of X and a subadditive potential \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F in \documentclass[12pt]{minimal}\begin{document}$C(X,\mathbb {R})$\end{document}C(X,R). By proving the upper semi-continuity and affinity of the entropy maps \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ(T,U∣Y) and \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h^+_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ+(T,U∣Y) on the space of all invariant Borel probability measures, we show that the relative local topological pressure with subadditive potentials determines the local measure-theoretic conditional entropies.

Observable optimal state points of subadditive potentials

For a sequence of subadditive potentials, a method of choosing state points with negative growth rates for an ergodicdynamical system was given in [5]. This paper firstgeneralizes this result to the non-ergodic dynamics, and thenproves that under some mild additional hypothesis, one can choosepoints with negative growth rates from a positive Lebesgue measureset, even if the system does not preserve any measure that isabsolutely continuous with respect to Lebesgue measure.