Shadowing, topological entropy and recurrence of induced Morse-Smale diffeomorphisms

2022/03/24 by Alexander Arbieto, Arbieto, Alexander, Jennyffer Bohorquez +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2203.13356

openalex publication_date 2022/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f : M → M be a Morse-Smale diffeomorphism defined on a compact and connected manifold without boundary. Let C(M) denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and C(f) : C(M) → C(M) denote the induced homeomorphism of f. We show in this paper that if M is the unit circle S¹ then the induced map C(f) has not the shadowing property. Also we show that the topological entropy of C(f) has only two possible values: 0 or ∞. In particular, we show that the entropy of C(f) is 0 when M is the unit circle S¹ and it is ∞ if the dimension of the manifold M is greater than two. Furthermore, we study the recurrence of the induced maps 2^f and C(f) and sufficient conditions to obtain infinite topological entropy in the hyperspace.

Cited by

• Polynomial entropy of induced homeomorphisms on C_k(\mathbbS¹) and C_k([0,1])
• Polynomial entropy on the n-fold symmetric product and its suspension

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