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.

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 the transformations involved exhibit functional tied-down renewal mixing properties strengthening those of [AS19].

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 “weak Renyi measures” which are “doubling at 0”.

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.

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 q-stable Lévy-bridges (1 < q ≤ 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.

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.

Let H denote the two-dimensional hyperbolic space and φ (t ∈ IR) the geodesic flow on H × TT , where TT is the natural identification of directions. Throughout this note we work with the model of the Poincaré disk instead of the Poincaré upper half-plane (sometimes also called the Lobachevsky plane). So we consider H = {z ∈ CI : |z| < 1}. Let Γ be a discrete group of isometries of H, and let H/Γ denote the surface defined by Γ equipped with the metric induced by the hyperbolic metric ρ (see [Be]). The space of line elements of H/Γ is XΓ := (H/Γ)× TT = (H × TT )/Γ (also equipped with the induced metric) and the geodesic flow transformations on XΓ are defined by φΓΓ(ω) = Γφ (ω). We consider the measure m =hyperbolic area × normalised Lebesgue measure on H × TT and the corresponding induced measure mΓ on (H/Γ)× TT . For a compact surface the dynamical system (H/Γ, (φΓ)t∈IR) is an Anosov system ([An1], [An2]), the measure mΓ is finite and φΓ is a Bernoulli flow. This is proven by using the existence of expanding and contracting flow invariant foliations (also used by Anosov and Sinai to show that φΓ is a K-flow (cf. [An2])) and applying the Ornstein isomorphism theory (Ornstein, Weiss [O-W]). Here we are mainly interested in the non-compact case and our result holds for conservative geodesic flows φΓ. The following characterisation of these dynamical systems is given in the work of Hopf and Tsuji (cf. [Ho1], [Ho2], [Ts1] and [Ts2]), which uses methods from potential theory.

In this note we show existence of bounded, continuous, transitive cocycles over a transitive action by homeomorphisms of any finitely generated group on a Polish space, and bounded, measurable, ergodic cocycles over any ergodic, probability-preserving action of ℤd.

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.

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 asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.

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.

In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable- and $\alpha$-rationally ergodic tranformations are presented.

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 consider the latter phenomenon and characterize it among transformations admitting generalized recurrent events.

We prove local limit theorems for fibred semiflows in both the Gaussian and non-Gaussian stable case.

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.

We discuss multiple versions of rational ergodicity and rational weak mixing for ‘nice’ transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.

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.

Abstract We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential $\chi ^2$form.

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.

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.