Specification properties for non-autonomous discrete systems

TL;DR

This paper introduces and analyzes strong and quasi-weak specification properties for non-autonomous discrete systems, demonstrating their dynamical significance, preservation under products, and implications for chaos and induced systems.

Contribution

It defines new specification properties for non-autonomous systems, proves their dynamical invariance, and explores their implications for chaos and induced systems.

Findings

Weak specification implies Devaney chaos in periodic systems.

Strong specification guarantees chaos in general systems.

Specification properties are preserved under finite products.

Abstract

In this paper notions of strong specification property and quasi-weak specification property for non-autonomous discrete systems are introduced and studied. It is shown that these properties are dynamical properties and are preserved under finite product. It is proved that a k-periodic non-autonomous system on intervals having weak specification is Devaney chaotic whereas if the system has strong specification then the result is true in general. Specification properties of induced systems on hyperspaces and probability measures spaces are also studied. Examples/counter examples are provided wherever necessary to support results obtained.