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Jon Aaronson

Publications and source records attributed to Jon Aaronson.

10 recordsLinked to original sources
arXiv2026

Dual mixing and quasi-mixing of inner function restrictions

Dual quasi-mixing and dual mixing of a measure preserving transformation are uniform individual ratio limit properties of the transfer operator which entail the quasi-mixing and mixing properties of Krickeberg (respectively). The real restriction of a normalised, parabolic inner function of the upper half plane preserves Lebesgue measure and is dual mixing iff it is exact. Certain other restrictions are ``singular'' dual quasi-mixing.

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arXiv2024

Extravagance, irrationality and Diophantine approximation

For an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotient function is integrable. We show that with respect to a ``continued fraction mixing'' measure for the Gauss map with the log of the partial quotient function non-integrable, almost all numbers are Liouville. We also exhibit Gauss-invariant, ergodic measures with arbitrary irrationality exponent. The proofs are applications of our study of the ``extravagance'' of positive, stationary, stochastic processes. In addition, we prove a Khinchin-type dichotomy for Diophantine approximation with respect to ergodic``weak Renyi measures'' which are ``doubling at $0$''.

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arXiv2023

Dynamics of inner functions revisited

We study the circle restrictions of inner functions of the unit disc showing that the local invertibility of a restriction is independent of its singularity set and proving a local characterization of analytic conditional expectations. We establish central limit properties for some stochastic processes driven by probability preserving restrictions via spectral analysis of their perturbed transfer operators.

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arXiv2012

Relative complexity of random walks in random sceneries

Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the α-stable CLT ($1<α\le2$). The results give invariants for relative isomorphism of these.

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arXiv2009

Predictability, entropy and information of infinite transformations

We show that a certain type of quasi finite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasi finite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with square root normalization. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most 1/2.

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arXiv2006

On the mixing coefficients of piecewise monotonic maps

We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse $ϕ$-mixing. If its invariant density is uniformly bounded away from 0, it is $ϕ$-mixing iff all images of all orders are big in which case it is $ψ$-mixing. Among $\b$-transformations, non-$ϕ$-mixing is generic. In this sense, the asymmetry of $ϕ$-mixing is natural.

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arXiv2004

Occupation times of sets of infinite measure for ergodic transformations

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.

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