Periodic Points of Asymmetric Bernoulli Shifts

It is well-known that Sharkovskii’s theorem gives a complete structure of periodic order for a continuous self-map on a closed bounded interval. As a further study, a natural problem is how to determine the location and number of periodic points for a specific map. This paper considers the periodic points of asymmetric Bernoulli shift, which is a piecewise linear chaotic map.

Harmonic models and Bernoullicity

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

Entropy of MV-algebraic dynamical systems: An example

We give examples showing that the Kolmogorov-Sinai entropy generator theorem is false for both upper and lower Riesz entropy of MV-algebraic dynamical systems, both two sided (i.e., analogous to two sided Bernoulli shifts) and one sided (i.e., analogous to one sided Bernoulli shifts).

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

Continuity of Lyapunov exponents for random two-dimensional matrices

The Lyapunov exponents of locally constant$\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.