Inheritance of minimality from continuous to discrete time systems

Roman Hric; L'ubomír Snoha

doi:10.3934/dcds.2024083

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2024, Volume 44, Issue 12: 3962-3979. Doi: 10.3934/dcds.2024083

This issue Previous Article Conservation laws and Hamilton-Jacobi equations on a junction: The convex case Next Article

Inheritance of minimality from continuous to discrete time systems

Roman Hric^, and
L'ubomír Snoha

Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40,974 01 Banská Bystrica, Slovakia

^*Corresponding author: Roman Hric
^*Corresponding author: Roman Hric

Received: November 2023

Revised: June 2024

Early access: June 2024

Published: December 2024

The authors are supported by VEGA grant 1/0634/24.

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Abstract

The paper is concerned with inheritance of density of orbits from a continuous time system to its time discretizations. We prove that for a two-sided minimal flow on a rather general topological space, there is a residual set of times such that, for every such time $ t $, the corresponding time-$ t $ homeomorphism inherits two-sided minimality. An analogous result is proved for one-sided minimal semiflows.

Apart from trivial implications, in noncompact spaces there is no relation between two-sided, forward, and backward minimality of flows and homeomorphisms. This is not difficult to demonstrate for flows by appropriate examples. It has not been known how to provide corresponding examples of homeomorphisms. Using the results mentioned above, we are able to do so.

As another application, we prove a continuous time analogue of Gottschalk's Theorem on spaces which do not admit minimal maps.

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Mathematics Subject Classification: Primary: 37B05, 37B20; Secondary: 37B02.

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