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Hanna Oppelmayer

Publications and source records attributed to Hanna Oppelmayer.

9 recordsLinked to original sources
arXiv2026

Relative stationary dynamical systems

Let $G$ be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure $μ$. We generalize the notion of $μ$-stationary systems to $μ$-stationary $G$-factor maps $π: (X,ν)\to (Y,η)$. For these stationary relations between dynamical systems, we provide a structure theorem, which generalizes the structure theorem of Furstenberg-Glasner. Furthermore, we show the existence and uniqueness of a relative version of the Poisson boundary in this setup.

math.DS↗
arXiv2026

Cone-Additive Functions for Random Walks on Free Products of Graphs

We define cone-additive functions for random walks on free products of countable graphs. These functions satisfy a limit theorem under mild assumptions. In fact, cone-additivity is present in several well-studied notions, like entropy, asymptotic range and drift. Cone-additivity can be seen as a separation property by space -- a quite different perspective than the well-studied concept of sub-additivity in the context of free products of groups, which is a separation by time. In our inhomogeneous setting of free products of graphs, this separation by space allows us to deduce new limit theorems for travelling salesman problems (that is, distance functions of lamplighter random walks on free products), for weight functions on edges and the range of the $r$-th visit.

math.PR↗
arXiv2024

On the amenable subalgebras of group von Neumann algebras

We approach the study of sub-von Neumann algebras of the group von Neumann algebra $LΓ$ for countable groups $Γ$ from a dynamical perspective. It is shown that $L(Γ)$ admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of sub-algebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant sub-algebra.

math.OA↗
arXiv2024

Unique ergodicity for random noninvertible maps on an interval

In this short note, we investigate non-invertible stochastic dynamical systems on the unit interval $[0, 1]$. We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.

math.DS↗
arXiv2021

Boundary entropy spectra as finite subsums

In this paper we provide a concrete construction of Furstenberg entropy values of $τ$-boundaries of the group $\mathbb{Z}[\frac{1}{p_1},\ldots,\frac{1}{p_{l}}]\rtimes \{p_1^{n_1}\cdots p_{l}^{n_{l}} \, : \, n_i\in\mathbb{Z}\}$ by choosing an appropriate random walk $τ$. We show that the boundary entropy spectrum can be realized as the subsum-set for any given finite sequence of positive numbers.

math.DS↗
arXiv2020

Random walks on dense subgroups of locally compact groups

Let $Γ$ be a countable discrete group, $H$ a lcsc totally disconnected group and $ρ: Γ\rightarrow H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup $L < H$ and bi-$L$-invariant probability measure $θ$ on $H$, a Furstenberg discretization $τ$ of $θ$ such that the Poisson boundary of $(H,θ)$ is a $τ$-boundary. Among other things, this technique allows us to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not $L^p$-irreducible for any $p \geq 1$, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group $Γ$ and two spread-out probability measures $τ_1$ and $τ_2$ on $Γ$ such that the boundary entropy spectrum of $(Γ,τ_1)$ is an interval, while the boundary entropy spectrum of $(Γ,τ_2)$ is a Cantor set.

math.DS↗
arXiv2020

Kudo-Continuity Of Entropy Functionals

We study in this paper real-valued functions on the space of all sub-$σ$-algebras of a probability measure space, and introduce the notion of Kudo-continuity, which is an a priori strengthening of continuity with respect to strong convergence. We show that a large class of entropy functionals are Kudo-continuous. On the way, we establish upper and lower continuity of various entropy functions with respect to asymptotic second order stochastic domination, which should be of independent interest. An application to the study of entropy spectra of $μ$-boundaries associated to random walks on locally compact groups is given.

math.PR↗