Lightly chaotic functional envelopes

TL;DR

This paper introduces the concept of light chaos in dynamical systems, explores its relationship with classical chaos notions, and establishes a connection between system chaos and the chaos of associated functional envelopes.

Contribution

It defines light chaos using subbases, provides examples illustrating its relation to classical chaos, and proves that Devaney chaos in a system corresponds to light chaos in its functional envelope.

Findings

Light chaos is a significant new dynamical property.

Devaney chaos in a map is equivalent to light chaos in its functional envelope.

Examples clarify the relationship between light chaos and classical chaos notions.

Abstract

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Severalexamples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f : X ! X, where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic.