Topological mixing and uniquely ergodic systems

Topological Mixing and Uniquely Ergodic Systems

Every ergodic transformation (X, 7, :~,/z) has an isomorphic system (Y, U, ~, v) which is uniquely ergodic and topologically mixing.

A Uniquely Ergodic Cellular Automaton

We construct a one-dimensional uniquely ergodic cellular automaton which is not nilpotent. This automaton can perform asymptotically infinitely sparse computation, which nevertheless never disappears completely. The construction builds on the self-simulating automaton of Gács. We also prove related results of dynamical and computational nature, including the undecidability of unique ergodicity,...

Existence of Non-uniform Cocycles on Uniquely Ergodic Systems

Every Ergodic Measure Is Uniquely Maximizing

Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f : X → R, a measure μ ∈ Mφ is called f -maximizing if ∫ f dμ = sup{ ∫ f dm : m ∈Mφ}. It is shown that if μ is any ergodic measure in Mφ, then there exists a continuous function whose unique maximizing measure is μ. More generally, if E is a non-e...

Topological Weak Mixing and Quasi-bohr Systems

A minimal dynamical system (X,T ) is called quasi-Bohr if it is a nontrivial equicontinuous extension of a proximal system. We show that if (X,T ) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of (X, T ) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is: AG′ = G implies weak mixing.) The converse does ...