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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.17448 |
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| _version_ | 1866912776579973120 |
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| author | Iketake, Kazutoyo |
| author_facet | Iketake, Kazutoyo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17448 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos Iketake, Kazutoyo Dynamical Systems Functional Analysis 47A16 (Primary) 37B10, 47B38 (Secondary) 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. |
| title | Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos |
| topic | Dynamical Systems Functional Analysis 47A16 (Primary) 37B10, 47B38 (Secondary) |
| url | https://arxiv.org/abs/2512.17448 |