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

In this paper, we investigate the chaotic behavior of the differential operator $\frac{d}{dx}$ on the space of smooth functions $C^{\mathrm{\infty }}([a,b])$ equipped with the $L^p$-norm ($1\le p\le \mathrm{\infty }$). We explicitly construct a homeomorphism between a subset of $C^{\mathrm{\infty }}([a,b])$ 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 $C^{\mathrm{\infty }}([a,b])$. We also prove that the differential operator is chaotic on the entire space $C^{\mathrm{\infty }}([a,b])$ using a similar approach.