Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0406580

Occupation times of sets of infinite measure for ergodic transformations

Jon Aaronson·Maximilian Thaler·Roland Zweimueller
View original source ↗Read source document ↗Export metadata ↓

Abstract

Assume that $T$ is a conservative ergodic measure preserving transformation of the infinite measure space $(X,\mathcal{A},μ)$.We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling-Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with c.f.-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These abstract results are illustrated by applications to interval maps with indifferent fixed points.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jon Aaronson, Maximilian Thaler, Roland Zweimueller. 2004-06-28. Occupation times of sets of infinite measure for ergodic transformations. https://arxiv.org/abs/math/0406580

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Monotonicity, global symplectification and the stability of Dry Ten Martini Problem

We prove that, for every irrational frequency and every analytic Type I potential, each supercritical spectral energy satisfying the gap-labelling condition is an endpoint of an open spectral gap. This establishes the conjecture of Ge--Jitomirskaya--You \cite{GJY,You} in the supercritical regime. Consequently, the ``all gaps open'' property of the supercritical almost Mathieu operator persists under sufficiently small analytic perturbations. The main ingredient is a global symplectification of the center bundle that preserves quantitative monotonicity. This allows us to study gap opening through the center dynamics of the dual long-range operator, which has no natural Schrödinger form. We first establish the result for trigonometric polynomial potentials and then pass to general analytic potentials by controlling the dependence on the truncation dimension. The proof combines a discrete Hellmann--Feynman identity, dimension-free Aubry duality in weighted analytic norms, and a quantitative cone argument based on pre-monotonicity. These estimates ensure that the gaps survive in the analytic limit. Our results establish analytic stability of the Dry Ten Martini Problem in the supercritical regime and give a partial answer to a question of M. Shamis on the persistence of periodic spectral gaps.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms via bounded solutions

We introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter $s$ in the critical strip. We identify sufficient conditions on the non-homogeneous term that induce a structural asymmetry between the solutions corresponding to the parameters $s$ and $1-s$. More precisely, if both solutions with initial value $1$ are bounded on $[1,\infty)$, then necessarily $\Re(s)=\tfrac12$. The initial condition associated with the unique bounded solution corresponding to a parameter $s$ represents a zero of the Mellin transform associated with the non-homogeneous term at the point $s$.

math.DS↗
arXiv2026

Self-similar Delone sets and Pisot numbers

We consider Delone point patterns with self-similarity. Under mild conditions, the similarity factor is a Pisot number if and only if the pattern is uniformly discrete. The classical case is a Meyer set $Λ$ with $Λ\supset θΛ$ for some $θ>1,$ for which $θ$ must be a Pisot number or a Salem number. When $Λ$ contains several similar copies of itself, the case of a Salem number drops out for $θ<2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.

math.DS↗