Kyewon Koh Park

On directional entropy functions

AbstractGiven aZ2-process, the measure theoretic directional entropy function,h(

The recurrence time for ergodic systems with infinite invariant measures

\vec v

A counterexample of the entropy of a skew product

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Spatial determinism for a free Z2-action

S^1 = \left\{ {\vec v:\left\| {\vec v} \right\| = 1} \right\} \subset R^2

TOPOLOGICAL DISJOINTNESS FROM ENTROPY ZERO SYSTEMS

% MathType!End!2!1!. We relate the directional entropy of aZ2-process to itsR2 suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ2-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aGδ subset ofS1. We study examples to investigate some properties of directional entropy functions.

The first return time properties of an irrational rotation

We introduce the notion of topological entropy dimension to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential, growth rate of orbits. We also introduce the dimension set, D(X,T ) ⊂ [0, 1], of a topological dynamical system to study the complexity of its factors. We construct a minimal example whose dimension set consists of one number. This implies the property that every nontrivial open cover has the same entropy dimension. This notion for zero entropy systems corresponds to the K-mixing property in measurable dynamics and to the uniformly positive entropy in topological dynamics for positive entropy systems. Using the entropy dimension, we are able to discuss the disjointness between the entropy zero systems. Properties of entropy generating sequences and their dimensions have been investigated.