Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.

Exploding Markov operators

A special class of doubly stochastic (Markov) operators is constructed. In a sense these operators come from measure preserving transformations and inherit some of their properties, namely ergodicity and positivity of entropy, yet they may have no pointwise factors.

Refined scales of weak-mixing dynamical systems: typical behaviour

We study sets of measure-preserving transformations on Lebesgue spaces with continuous measures taking into account extreme scales of variations of weak mixing. It is shown that the generic dynamical behaviour depends on subsequences of time going to infinity. We also present corresponding generic sets of (probability) invariant measures with respect to topological shifts over finite alphabets and Axiom A diffeomorphisms over topologically mixing basic sets.

Ergodicity Space for Measure-Preserving Transformations

We introduce the concept of ergodicity space of a measure-preserving transformation and will present some of its properties as an algebraic weight for measuring the size of the ergodicity of a measure-preserving transformation. We will also prove the invariance of the ergodicity space under conjugacy of dynamical systems.

Conditions for rational weak mixing

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.

On multiple recurrence and other properties of ‘nice’ infinite measure-preserving transformations

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.

Return- and hitting-time limits for rare events of null-recurrent Markov maps

We determine limit distributions for return- and hitting-time functions of certain asymptotically rare events for conservative ergodic infinite measure preserving transformations with regularly varying asymptotic type. Our abstract result applies, in particular, to shrinking cylinders around typical points of null-recurrent renewal shifts and infinite measure preserving interval maps with neutral fixed points.