Systems with Almost Specification Property May Have Zero Entropy

TL;DR

This paper demonstrates that systems with the almost specification property can have zero entropy, highlighting a key difference from systems with the stronger specification property which always have positive entropy.

Contribution

It establishes the existence of systems with almost specification property and zero entropy, and provides conditions for positive entropy.

Findings

Systems with almost specification can have zero entropy.

Systems with the specification property must have positive entropy.

A necessary and sufficient condition for positive entropy is identified.

Abstract

It is shown that there exist systems having almost specification property and zero entropy. Since Sigmund has shown that systems with specification property must have positive entropy, this result reveals further the difference between almost specification and specification. Moreover, one can step on to obtain a both sufficient and necessary condition to ensure positive entropy.