Linear repetitivity and subadditive ergodic theorems for cut and project sets

Uniform ergodic theorems on subshifts over a finite alphabet

We investigate uniform ergodic type theorems for additive and subadditive functions on a subshift over a finite alphabet. We show that every strictly ergodic subshift admits a uniform ergodic theorem for Banach-space-valued additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides in particular a sufficient condition for unique ergodicity.

Kingman次可加遍历定理的间隙隙推广

A gapped generalization of Kingman's subadditive ergodic theorem

本文推广了Kingman次可加遍历定理至间隙隙次几乎次可加性条件，并应用于单向平移上适当解耦测度的相对熵存在性证明。

Dynamical properties of almost repetitive Delone sets

We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense…

Ratio ergodic theorems: From Hopf to Birkhoff and Kingman

Hopf's ratio ergodic theorem has an inherent symmetry which we exploit to provide a simplification of standard proofs of Hopf's and Birkhoff's ergodic theorems. We also present a ratio ergodic theorem for conservative transformations on a $σ$-finite measure space, generalizing Kingman's ergodic theorem for subadditive sequences and generalizing previous results by Akcoglu and Sucheston.

不变容度的遍历性与混合性及其应用

Ergodicity and Mixing of invariant capacities and applications

该研究引入公共条件期望与骨架概率方法，研究不变上概率的遍历定理与弱混合性，给出多个应用与实例，并推广至经典遍历理论与非不变概率场景。

Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets' multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work represents a significantly simplified means of obtaining the result, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman's subadditive ergodic theorem.

Measures of full dimension on self-affine sets

Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we show that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself.

Mather sets for sequences of matrices and applications to the study of joint spectral radii

The joint spectral radius of a compact set of d-times-d matrices is defined ?to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article we investigate the ergodic-theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise ma…

随机动力学中的极值李雅普诺夫指数

Extremal Lyapunov exponents in random and simultaneous dynamics

针对光滑紧测度空间上含至少一个度严格大于1映射的有限映射集合生成的随机动力学，基于Kingman遍历定理（Birkhoff遍历定理的推广）推导得到刻画典型轨道不稳定性的极值李雅普诺夫指数表达式。

大部分收缩的随机映射

Mostly contracting random maps

研究紧度量空间上Lipschitz变换的独立随机迭代的长期行为，引入Lyapunov指数概念，证明该类映射在适当拓扑下是开的，并满足大数定律、Palis全局猜想等性质。

Repetitive Delone Sets and Quasicrystals

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is called repetitive if there is a function M(T) such that every ball of radius M(T)+T contains a copy of each kind of patch of radius T that occurs in the set. This is equivalent to the minimality of an associated topological dynamical system with R^n-action. There is a lower bound for M(T) in terms of N(T), namely N(T) = O(M(T)^n), but no general upper bound. T…