On the topological pressure of random bundle transformations in sub-additive case

Sub-additive unstable topological pressure on saturated subsets

In this paper, we continue our investigation on sub-additive pressures for $C^1$-smooth partially hyperbolic diffeomorphisms. Under the assumption of unstable almost product property, we show that the unstable Bowen topological pressure on the saturated set of a given non-empty compact connected set of invariant measures equals the infimum of the summation of unstable metric entropy and the corresponding \emph{Lyapunov exponent}, where the infimum is taken over all invariant measures inside the compact connected set above. Moreover, we also show that the unstable topological capacity pressure …

The Asymptotically Additive Topological Pressure: Variational Principle For Non Compact and Intersection of Irregular Sets

Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space and $f:X\rightarrow X$ is a continuous map. Using the concepts of \textit{g-almost product property} and \textit{uniform separation property} introduced by Pfister and Sullivan in \cite{Pfister2007}, we give a variational principle for certain non-compact with relation the asymptotically additive topological pressure. We also study the set of points that are irregular for an collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure…

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. For any ergodic measure, we show that this unstable metric pressure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponents with respect to the ergodic measure. On the other hand, we also define unstable topological and metric pressure in terms of the Bowen's picture and the capacity picture. We show that all these definitions of unstable metric pressure actually coincide for any ergodic measure.

非加性中和Bowen拓扑压的变分原理

Variational principle for non-additive neutralized Bowen topological pressure

本研究将中和Bowen拓扑熵的变分原理推广至非加性情形，针对具缓增畸变的非加性势，建立了非加性中和Bowen拓扑压的变分原理。

Variational principle for subadditive sequence of potentials in bundle RDS

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

Pre-image Variational Principle for Bundle Random Dynamical Systems

The pre-image topological pressure is defined for bundle random dynamical systems. A variational principle for it has also been given.

关于与某些相对压力函数相关的近可加序列的Gibbs测度

On Gibbs measures for almost additive sequences associated to some relative pressure functions

研究了与某些相对压力函数相关的近可加序列的Gibbs测度性质及其在因子映射中的应用。

Almost-additive thermodynamic formalism for countable Markov shifts

We introduce a definition of pressure for almost-additive sequences of continuous functions defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs and equilibrium measures. Applications are given to the study of maximal Lypaunov exponents of product of matrices.

Lyapunov spectrum of asymptotically sub-additive potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-a…

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman's sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs measures.

Topological pressure, mistake functions and average metric

In this paper, we showed that the Pesin pressure of any subset under a mistake function is equal to the classical Pesin pressure of the subset in dynamical systems. Our result extended the result of [1] in additive case, which proved the topological pressure of the whole system is self adaptable under a mistake function. As an application, we showed that the Pesin pressure defined by average metric is equal to the classical Pesin pressure.

基于不稳定压强的双曲动力学盒维数上界

An upper bound for the box dimension of the hyperbolic dynamics via unstable pressure

该研究利用次可加不稳定压强推导C¹双曲集上盒维数的上界，同时给出拓扑压强的新表达式，丰富了双曲动力学相关理论。

Pressures for geodesic flows of rank one manifolds

We study the geodesic flow on the unit tangent bundle of a rank one manifold and we give conditions under which all classical definitions of pressure of a Hölder continuous potential coincide. We provide a large deviation statement, which allows to neglect (periodic) orbits that lack sufficient hyperbolic behavior. Our results involve conditions on the potential, that take into consideration its properties in the nonhyperbolic part of the manifold. We draw some conclusions for the construction of equilibrium states.

Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone

The sub-additive pressure function $P(s)$ for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution $s_0$ to $P(s_0)=1$, were introduced by K. Falconer in his seminal 1988 paper on self-affine fractals. The affinity dimension prescribes a value for the Hausdorff dimension of a self-affine set which is known to be correct in generic cases and in an increasing range of explicit cases. It was shown by Feng and Shmerkin in 2014 that the affinity dimension depends continuously on the IFS. In this article we prove that when the linear parts of the affini…

相对熵、拓扑压力与局部紧致索菲克群作用的变分原理

Relative entropy, topological pressure and variational principle for locally compact sofic group actions

本文研究了局部紧致索菲克群作用的相对熵和拓扑压力，建立了变分原理，并给出了有向测度的充分条件。

Additive, almost additive and asymptotically additive potential sequences are equivalent

Motivated by various applications and examples, the standard notion of potential for dynamical systems has been generalized to almost additive and asymptotically additive potential sequences, and the corresponding thermodynamic formalism, dimension theory and large deviations theory have been extensively studied in the recent years. In this paper, we show that every such potential sequence is actually equivalent to a standard (additive) potential in the sense that there exists a continuous potential with the same topological pressure, equilibrium states, variational principle, weak Gibbs measu…

Local Entropy Theory of a Random Dynamical System

In this paper we extend the notion of a continuous bundle random dynamical system to the setting where the action of $\R$ or $\N$ is replaced by the action of an infinite countable discrete amenable group. Given such a system, and a monotone sub-additive invariant family of random continuous functions, we introduce the concept of local fiber topological pressure and establish an associated variational principle, relating it to measure-theoretic entropy. We also discuss some variants of this variational principle. We introduce both topological and measure-theoretic entropy tuples for continuous…

几乎可加势的平衡态、零温度极限与熵连续性

Equilibrium States, Zero Temperature Limits and Entropy Continuity for Almost-Additive Potentials

本文研究了非紧致拓扑混合可数 Markov 移位上几乎可加势的平衡态，证明了唯一平衡态的存在性、Gurevich 压力的可微性与熵连续性，并将零温度极限与熵连续性结论推广至有限本原情形之外。

On factors of Gibbs measures for almost additive potentials

Let $(X, σ_X), (Y, σ_Y)$ be one-sided subshifts with the specification property and $π:X\rightarrow Y$ a factor map. Let $μ$ be a unique invariant Gibbs measure for a sequence of continuous functions $\F=\{\log f_n\}_{n=1}^{\infty}$ on $X$, which is an almost additive potential with bounded variation. We show that $πμ$ is also a unique invariant Gibbs measure for a sequence of continuous functions $\G=\{\log g_n\}_{n=1}^{\infty}$ on $Y$. When $(X, σ_X)$ is a full shift, we characterize $\G$ and $μ$ by using relative pressure. This almost additive potential $\G$ is a generalization of a continu…

Non-conformal repellers and the continuity of pressure for matrix cocycles

The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated …

Weighted thermodynamic formalism and applications

Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $Φ$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $Φ$. We setup the weighted variational principle. In the case that $X, Y$ are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional wei…