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Jaworski-type embedding theorems of one-sided dynamical systems

Volume 253 / 2021

Hisao Kato Fundamenta Mathematicae 253 (2021), 205-218 MSC: Primary 37A35, 37C45, 54H20; Secondary 54F45, 37B45, 37M10. DOI: 10.4064/fm894-6-2020 Published online: 18 December 2020

Abstract

Embeddings of two-sided dynamical systems in the two-sided shift $(\mathbb R ^{\mathbb Z },\sigma )$ of the real line $\mathbb R $ have been studied by many authors. In this paper, we introduce the notion of trajectory-embedding for the one-sided shift $(\mathbb R ^{\mathbb N },\sigma )$ and we give trajectory-embedding theorems for some one-sided dynamical systems. In particular, we show that if $X$ is a finite-dimensional compact metric space and $T:X\to X$ is a doubly 0-dimensional map with at most 0-dimensional set ${\rm P} (T)$ of periodic points (i.e., $\dim {\rm P} (T)\leq 0$), then there is a trajectory-embedding of $(X,T)$ in $(\mathbb R ^{\mathbb N },\sigma )$. We study such embedding theorems of one-sided dynamical systems from a different perspective than Jaworski–Nerurkar–Gutman.

Authors

  • Hisao KatoInstitute of Mathematics
    University of Tsukuba
    Tsukuba, Ibaraki 305-8571, Japan
    e-mail

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