Closed relations with non-zero entropy that generate no periodic points

Iztok Banič; Goran Erceg; Judy Kennedy

doi:10.3934/dcds.2022089

Article Contents

2022, Volume 42, Issue 10: 5137-5166. Doi: 10.3934/dcds.2022089

This issue Previous Article Ground states for a system of nonlinear Schrödinger equations with singular potentials Next Article

Closed relations with non-zero entropy that generate no periodic points

1.
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000 Maribor
2.
Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana
3.
Andrej Marušič Institute, University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia
4.
Faculty of Science, University of Split, Rudera Boškovića 33, Split, Croatia
5.
Department of Mathematics, Lamar University, 200 Lucas Building, P.O. Box 10047, Beaumont, TX 77710, USA

^*Corresponding author: Goran Erceg
^*Corresponding author: Goran Erceg

Received: February 28, 2022

Early access: July 2022

Published: September 2022

This work was supported by the Slovenian Research Agency under the research program P1-0285

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Abstract

The paper is motivated by E. Akin's book about dynamical systems and closed relations [1], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [9].

In present paper, we introduce the entropy of a closed relation $ G $ on any compact metric space $ X $ and show its basic properties. We also introduce when such a relation $ G $ generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation $ G $ on $ [0,1] $ to have a non-zero entropy. Then we present various examples of closed relations $ G $ on $ [0,1] $ such that (1) the entropy of $ G $ is non-zero, (2) no periodic point or exactly one periodic point is generated by $ G $, and (3) no Cantor set is finitely generated by $ G $.

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Mathematics Subject Classification: Primary: 37B40, 37B45, 37E05; Secondary: 54C08, 54E45, 54F15, 54F17.

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Figure 1. The sets $ L $ and $ R $

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Figure 2. The relation $ H_{a,b} $

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Figure 3. The relation $ H $ from Theorem 7.2

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Figure 4. The relation $ H $ from Theorem 7.3

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Figure 5. The relation $ H $ from Lemma 7.7

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Figure 6. The relation $ H $ from Theorem 7.9

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