arXiv subjects

Elias Zimmermann

Publications and source records attributed to Elias Zimmermann.

4 recordsLinked to original sources

arXiv2025

Mixing and equipartition for automorphism invariant processes on regular trees

The paper is devoted to equipartition of measured information for finite state processes over regular trees whose laws are invariant under all parity preserving tree automorphisms. We show almost everywhere equipartition for ergodic processes along spheres and balls in every horosphere. Moreover, under a quantitive mixing condition we obtain a Shannon-McMillan-Breiman theorem along metric spheres of even radius.

math.DS↗

arXiv2025

Exponential phi-mixing implies exponential psi-mixing for Markov fields on bounded degree graphs

We show that for non-degenerate $k$-Markovian random fields with finite state space over a bounded degree graph with exponential growth rate $\theta$ uniform $\phi$-mixing with exponential decay rate $\lambda > 3\theta$ implies uniform $\psi$-mixing with exponential decay rate $(\lambda - 3\theta)/9$. As an application we obtain exponential $\psi$-mixing for Gibbs fields on regular trees arising from finite range potentials such as the Ising model at low inverse temperature or the Potts model with sufficiently many spin states.

math.PR↗

arXiv2022

Fiber entropy and algorithmic complexity of random orbits

Let $Θ$ be a finite alphabet. We consider a bundle of measure preserving transformations $(T_θ)_{θ\in Θ}$ acting on a probability space $(X,μ)$, which are chosen randomly according to an ergodic stochastic process $(Ξ,ν,σ)$ with state space $Θ$. This describes a paradigmatic case of a random dynamical system (RDS). Considering a finite partition $\mathcal{P}$ of $X$ we show that the conditional algorithmic complexity of a random orbit $x, T_{α_{0}}(x),T_{α_{1}}\circ T_{α_{0}}(x),...$ in $X$ along a sequence $α= α_{0}α_{1}α_{2}...$ in $Ξ$ equals almost surely the fiber entropy of the RDS with respect to $\mathcal{P}$, whenever the latter is ergodic. This extends a classical result of A. A. Brudno connecting algorithmic complexity and entropy in deterministic dynamical systems.

math.DS↗

arXiv2022

Strict irreducibility of Markov chains and ergodicity of skew products

We consider a family of measure preserving transformations, which act on a common probability space and are chosen at random by a stationary ergodic Markov chain. This setting defines an instance of a random dynamical system (RDS), which may be described in terms of a step skew product. In many contexts it is desirable to know whether ergodicity of the family implies ergodicity of the skew product. Introducing the notion of strict irreducibility for Markov kernels we shall characterize the class of Markov chains for which the aforementioned implication holds true. We thereby extend a sufficient condition of Bufetov for the case of finite state Markov chains to general state spaces and show that it is in fact also necessary. As an application we obtain an explicit description of the limit in ergodic theorems for a suitable class of random transformations.

math.DS↗