arXiv subjects

Kosma Kasprzak

Publications and source records attributed to Kosma Kasprzak.

4 recordsLinked to original sources

arXiv2026

IC-rigidity for abelian actions

Let $G$ be a group acting continuously on a compact metric space $X$. This naturally induces an action of $G$ on the space $\mathcal{K}(X)$ of nonempty compact subsets of $X$. In [KS26], Kra and Schmieding define a system $(X,G)$ to be IC-rigid if the union of the supports of the $G$-invariant Borel probability measures on $\mathcal{K}(X)$ is as small as possible. They leave open the question of existence of IC-rigid abelian systems. We show the existence of IC-rigid actions for three classes of abelian groups: mixing rank-one $\mathbb{Z}$-actions, horocycle flows, and certain mixing actions on a Cantor space of $G=\bigoplus_{n=1}^{\infty}G_n$, where $(G_n)_n$ is any sequence of nontrivial finite abelian groups. In the course of proving these results, we introduce the notion of weakly asymptotic pairs and establish its close connection to topological minimal self-joinings.

math.DS↗

arXiv2026

The conjugacy and flip conjugacy problem for Cantor minimal systems

We prove that topological conjugacy and flip conjugacy of minimal homeomorphisms of a Cantor space are complete analytic relations and hence are not Borel. We also prove that mutual reducibility by injective continuous graph homomorphisms and topological graph isomorphism are complete analytic relations on the space of nonempty compact graphs on a fixed Cantor vertex space with continuous chromatic number two. Both relations remain complete analytic on the larger space of nonempty compact graphs on the same Cantor vertex space with continuous chromatic number two or three.

math.DS↗

arXiv2025

Distribution of polynomial orbits in Toeplitz systems

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study density of the polynomial orbits in Toeplitz systems -- we show that it implies equidistribution of the polynomial orbits in the class of regular Toeplitz systems, but not in the class of strictly ergodic ones.

math.DS↗

arXiv2025

Equidistribution of orbits at polynomial times in rigid dynamical systems

We study distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems $(X, T)$. First, we prove that if there exists an increasing sequence $(q_n)$ for which the rigidity condition \[ \max_{t 1$ a much weaker rigidity condition \[ \max_{t<q_n^{C-1}}\sup\limits_{x\in X}d\left(x, T^{tq_n}x\right)=o(1) \] implies density of all orbits $(T^{n^C}x)$ in totally uniquely ergodic systems, as long as the sequence $(ω(q_n))$ is bounded.

math.DS↗