Jon Aaronson
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View article: Extravagance, irrationality and Diophantine approximation
Extravagance, irrationality and Diophantine approximation Open
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 wi…
View article: Generalized uniform laws for tied-down occupation times of infinite ergodic transformations
Generalized uniform laws for tied-down occupation times of infinite ergodic transformations Open
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 ti…
View article: Dynamics of inner functions revisited
Dynamics of inner functions revisited Open
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 …
View article: Functional limits for "tied down" occupation time processes of infinite ergodic transformations
Functional limits for "tied down" occupation time processes of infinite ergodic transformations Open
We prove functional, distributional limit theorems for the occupation times of pointwise dual ergodic transformations at "tied-down" times immediately after "excursions". The limiting processes are tied down Mittag-Leffler processes and th…
View article: Local limit theorems for suspended semiflows
Local limit theorems for suspended semiflows Open
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 va…
View article: Tied-down occupation times of infinite ergodic transformations
Tied-down occupation times of infinite ergodic transformations Open
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 variable…
View article: Rational ergodicity of step function skew products
Rational ergodicity of step function skew products Open
We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when the rotation number is a quadratic irrational. The latter arises from a consideration of the asympto…
View article: Rational ergodicity of Step function Skew Products
Rational ergodicity of Step function Skew Products Open
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 …
View article: Symmetric Birkhoff sums in infinite ergodic theory
Symmetric Birkhoff sums in infinite ergodic theory Open
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. Also, we cons…
View article: Distributional limits of positive, ergodic stationary processes and\n infinite ergodic transformations
Distributional limits of positive, ergodic stationary processes and\n infinite ergodic transformations Open
In this note we identify the distributional limits of non-negative, ergodic\nstationary processes, showing that all are possible.\n Consequences for infinite ergodic theory are also explored and new examples\nof distributionally stable- an…
View article: Conditions for rational weak mixing
Conditions for rational weak mixing Open
We exhibit rationally ergodic, spectrally weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.