Topological entropy of fixed-point free flows

Thomas, Romeo F.

doi:10.1090/S0002-9947-1990-1010414-5

Topological entropy of fixed-point free flows
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by Romeo F. Thomas

Trans. Amer. Math. Soc. 319 (1990), 601-618

DOI: https://doi.org/10.1090/S0002-9947-1990-1010414-5

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Abstract:

Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen’s definition. Also we show here that this definition is equivalent to Bowen’s definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.

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