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arXiv · 2512.17448

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

Kazutoyo Iketake
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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.

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Kazutoyo Iketake. 2025-12-19. Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos. https://arxiv.org/abs/2512.17448

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