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

  • *Corresponding author: Goran Erceg

    *Corresponding author: Goran Erceg 

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

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  • 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 $.

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


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

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

    Figure 3.  The relation $ H $ from Theorem 7.2

    Figure 4.  The relation $ H $ from Theorem 7.3

    Figure 5.  The relation $ H $ from Lemma 7.7

    Figure 6.  The relation $ H $ from Theorem 7.9

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