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Jon. Aaronson

Publications and source records attributed to Jon. Aaronson.

15 recordsLinked to original sources
arXiv2023

Generalized uniform laws for tied-down occupation times of infinite ergodic transformations

We establish a conditional limit theorem for occupation times of infinite ergodic transformations under a tied-down condition, that is, the condition that the orbit returns to a reference set with finite measure at the final observation time. The class of limit distributions is the generalization of the uniform distribution which was discovered by M. Barlow, J. Pitman and M. Yor in [S\'eminaire de Probabilit\'es XXIII. Lecture Notes in Mathematics, volume 1372 (1989), 294--314]. For the proof we utilize operator renewal theory. Our result can be applied to intermittent maps with two or more indifferent fixed points.

math.DS↗
arXiv2020

Local limit theorems for suspended semiflows

We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows of cyclic covers.

math.DS↗
arXiv2019

Tied-down occupation times of infinite ergodic transformations

We prove distributional limit theorems (conditional and integrated) for the occupation times of certain weakly mixing, pointwise dual ergodic transformations at "tied-down" times immediately after "excursions". The limiting random variables include the local times of $p$-stable L\'evy-bridges ($1<p\le 2$) and the transformations involved exhibit "tied-down renewal mixing" properties which refine rational weak mixing. Periodic local limit theorems for Gibbs-Markov and AFU maps are also established.

math.DS↗
arXiv2019

On the bounded cohomology of ergodic group actions

In this note we show existence of bounded, transitive cocycles over a transitive action of a finitely generated group, and bounded, ergodic cocycles over an ergodic, probability preserving action of $\Bbb Z^d$.

math.DS↗
arXiv2017

Rational ergodicity of Step function Skew Products

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.

math.DS↗
arXiv2017

Discrepancy Skew Products and Affine Random Walks

We prove bounded rational ergodicity for some discrepancy skew products whose rotation number has bad rational approximation. This is done by considering the asymptotics of associated affine random walks.

math.DS↗
arXiv2016

Symmetric Birkhoff sums in infinite ergodic theory

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.

math.DS↗
arXiv2013

Rational weak mixing in infinite measure spaces

Rational weak mixing is a measure theoretic version of Krickeberg's strong ratio mixing property for infinite measure preserving transformations. It requires "{\tt density}" ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey's strong ratio limit property. The power, subsequence version of the property is generic.

math.DS↗
arXiv2012

IP-rigidity and eigenvalue groups

We examine the class of increasing sequences of natural numbers which are IP-rigidity sequences for some weakly mixing probability preserving transformation. This property is closely related to the uncountability of the eigenvalue group of a corresponding non-singular transformation. We give examples, including a super-lacunary sequence which is not IP-rigid.

math.DS↗
arXiv2011

Limit theory for some positive, stationary processes with infinite mean

We prove distributional limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from non-integrable observables over certain piecewise expanding maps. This is done by extending Darling-Kac theory to a suitable family of infinite measure preserving transformations.

math.DS↗
arXiv2006

Exchangeable, Gibbs and equilibrium measures for Markov subshifts

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of "symmetric measure": exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.

math.PR↗
arXiv2004

A salad of cocycles

We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed. Smooth versions of some of the constructions are also given.

math.DS↗