arXiv subjects

Miriam Lorenzo-Laguno

Publications and source records attributed to Miriam Lorenzo-Laguno.

2 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

Lower bounds on mixing rates for a class of mixing flows on surfaces

We study mixing rates for locally Hamiltonian flows on compact surfaces with asymmetric logarithmic singularities. For a full measure set of such flows, we show that the decay of correlations of smooth observables cannot be uniformly faster than a power of $\log t$. In particular, there exist sequences of times and observables for which correlations admit lower bounds of order $(\log t)^{-2-ν}$ for any $ν>0$. We further show that for a typical Arnol'd flow on $\mathbb T^2$, the self-correlation of every box in the minimal component is bounded below by $(\log t)^{-1}$ along an unbounded sequence of times. Motivated by questions in spectral theory, we also construct examples of such flows for which the self-correlation of a box fails to be square-integrable. These results complement previous upper bounds for correlations in both settings, which are also of polynomial order in $\log t$, and show that logarithmic decay rates are essentially sharp along sequences of times.

math.DS↗