Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos

TL;DR

This paper demonstrates that the differential operator on smooth functions exhibits Devaney's chaos by constructing invariant sets and showing the operator's chaotic behavior on the entire space, linking it to symbolic dynamics.

Contribution

It explicitly constructs invariant sets in $C^ abla([a,b])$ where the differential operator behaves chaotically, extending the understanding of chaos in functional analysis.

Findings

Invariant sets densely configured in $C^ abla([a,b])$

Differential operator exhibits Devaney's chaos on the entire space

Homeomorphism with shift space established

Abstract

In this paper, we investigate the chaotic behavior of the differential operator $\frac{d}{dx}$ on the space of smooth functions $C^\infty([a,b])$ equipped with the $L^p$-norm ($1\le p\le\infty$). We explicitly construct a homeomorphism between a subset of $C^\infty([a,b])$ and the shift space. Moreover, inspired by symbolic dynamics, we demonstrate that invariant sets, on which the differential operator behaves analogously to the shift, are densely configured in $C^\infty([a,b])$. We also prove that the differential operator is chaotic on the entire space $C^\infty([a,b])$ using a similar approach.