Uniform (projective) hyperbolicity or no hyperbolicity : a dichotomy for generic conservative maps
Unique Bernoulli -measures
We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the -measure.
Universally finitary symbolic extensions
We prove that by considering a finitary (almost continuous) symbolic extension of a topological dynamical system instead of a continuous extension, one cannot achieve any drop of the entropy of the extension.
Using density matrices in classical dynamical systems
When the intrinsic algebraic entropy is not really intrinsic
The intrinsic algebraic entropy ent(ɸ) of an endomorphism ɸ of an Abelian group G can be computed using fully inert subgroups of ɸ-invariant sections of G, instead of the whole family of ɸ-inert subgroups. For a class of groups containing the groups of finite rank, aswell as those groupswhich are trajectories of finitely generated subgroups, it is proved that only fully inert subgroups of the group itself are needed to comput ent(ɸ). Examples show how the situation may be quite different outside...
Zero Krengel entropy does not kill Poisson entropy
We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann–Kakutani odometer), but whose associated Poisson suspension has positive entropy.