Uniform Generators, Symbolic Extensions with an Embedding, and Structure of Periodic Orbits

Abstract

For a topological dynamical system (X, T ) we define a uniform generator as a finite measurable partition such that the symmetric cylinder sets in the generated process shrink in diameter uniformly to zero. The problem of existence and optimal cardinality of uniform generators has lead us to new challenges in the theory of symbolic extensions. At the beginning we show that uniform generators can be identified with so-called symbolic extensions with an embedding, i.e., symbolic extensions admitting an equivariant measurable selector from preimages. From here we focus on such extensions and we strive to characterize the collection of the corresponding extension entropy functions on invariant measures. For aperiodic zero-dimensional systems we show that this collection coincides with that of extension entropy functions in arbitrary symbolic extensions, which, by the general theory of symbolic extensions, is known to coincide with the collection of all affine superenvelopes of the entropy structure of the system. In particular, we recover, after [Bu16], that an aperiodic zero-dimensional system is asymptotically hexpansive if and only if it admits an isomorphic symbolic extension. Next we pass to systems with periodic points, and we introduce the notion of a period tail structure, which captures the local growth rate of periodic orbits. Finally, we succeed in precisely identifying the wanted collection of extension entropy functions in symbolic extensions with an embedding: these are all the affine 1. There exists a uniform generator 2 P in (X, T ), of cardinality ;1 See Remark 1.3 for the precise meaning of measurability. 2 Later we will show that the measurability assumption of P can be dropped. That is, the existence of a "non-measurable uniform generator" implies the existence of a measurable one-see Remark 4.15.

Symbolic extensions and uniform generators for topological regular flows

“…A uniform generator is a Borel partition P of X such that the diameter of n k=−n T −k P goes to zero when n goes to infinity. The following statement follows from Theorem 1.2 in [10]. The characterization for clopen uniform generators was first proved in [18].…”

“…Formally such a code is given by a topological extension by a subshift over a finite alphabet, also called a symbolic extension. More recently T.Downarowicz and the author [10] have related the theory of symbolic extensions with a Krieger-like generators problem. For a discrete topological system (X, T ) they introduced uniform generators as Borel partitions P of X whose iterated partitions P [−n,n] T := n k=−n T k P have a diameter going to zero with n, in other terms sup y∈P [−n,n] T (x) d(y, x) goes to zero uniformly in x when n goes to infinity (with d being the distance on X).…”

“…For a discrete topological system (X, T ) they introduced uniform generators as Borel partitions P of X whose iterated partitions P [−n,n] T := n k=−n T k P have a diameter going to zero with n, in other terms sup y∈P [−n,n] T (x) d(y, x) goes to zero uniformly in x when n goes to infinity (with d being the distance on X). By Theorem 1 in [10] a uniform generator is given by a symbolic extension with a Borel embedding and vice versa. For aperiodic systems the existence of symbolic extensions is equivalent to the existence of uniform generators whereas the presence of periodic points generates other constraints to build uniform generators (see Theorem 55 in [10]).…”

“…By Theorem 1 in [10] a uniform generator is given by a symbolic extension with a Borel embedding and vice versa. For aperiodic systems the existence of symbolic extensions is equivalent to the existence of uniform generators whereas the presence of periodic points generates other constraints to build uniform generators (see Theorem 55 in [10]).…”

Topological and almost Borel universality for systems with the weak specification property

“…The map φ may be then extended equivariantly on the whole set lim ← −k Λ k (see Remark 14 of [BD16] for details).…”