Quenched Poisson processes for random subshifts of finite type

Longest common substring for random subshifts of finite type

In this paper, we study the behaviour of the longest common substring for random subshifts of finite type (for dynamicists) or of the longest common substring for random sequences in random environments (for probabilists). We prove that, under some exponential mixing assumptions, this behaviour is linked to the Rényi entropy of the stationary measure. We emphasize that what we establish is a quenched result.

Exponential law for random subshifts of finite type

In this paper we study the distribution of hitting times for a class of random dynamical systems. We prove that for invariant measures with super-polynomial decay of correlations hitting times to dynamically defined cylinders satisfy exponential distribution. Similar results are obtained for random expanding maps. We emphasize that what we establish is a quenched exponential law for hitting times.

Hitting times and periodicity in random dynamics

We prove quenched laws of hitting time statistics for random subshifts of finite type. In particular we prove a dichotomy between the law for periodic and for non-periodic points. We show that this applies to random Gibbs measures.

Return times at periodic points in random dynamics

We prove a quenched limiting law for random measures on subshifts at periodic points. We consider a family of measures $\{μ_ω\}_{ω\inΩ}$, where the `driving space' $Ω$ is equipped with a probability measure which is invariant under a transformation $θ$. We assume that the fibred measures $μ_ω$ satisfy a generalised invariance property and are $ψ$-mixing. We then show that for almost every $ω$ the return times to cylinders $A_n$ at periodic points are in the limit compound Poisson distributed for a parameter $\vartheta$ which is given by the escape rate at the periodic point.

随机动力系统通过全阶混合的淬火极限定理

Quenched limit theorems via mixing of all orders for random dynamical systems

将全阶混合概念推广到随机动力系统，并利用它建立淬火极限定理，包括中心极限定理和泊松极限定理，具有更弱的随机常数可积性条件并适用于可逆系统。

Poisson processes for subsystems of finite type in symbolic dynamics

Let $Δ\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an irreducible and aperiodic shift of finite type $(Σ_{A}^{+},§)$. Let $Σ_Δ$ be the subshift of allowable paths in the graph of $Σ_{A}^{+}$ which only passes through the vertices of $Δ$. For a random point $x$ chosen with respect to an equilibrium state $μ$ of a Hölder potential $ϕ$ on $Σ_{A}^{+}$, let $τ_{n}$ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $§^k x$ belong to $Δ$. We prove that this point process converges i…