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On fixed point theorems and nonsensitivity

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Abstract

Sensitivity is a prominent aspect of chaotic behavior of a dynamical system. We study the relevance of nonsensitivity to fixed point theory in affine dynamical systems. We prove a fixed point theorem which extends Ryll-Nardzewski’s theorem and some of its generalizations. Using the theory of hereditarily nonsensitive dynamical systems we establish left amenability of Asp(G), the algebra of Asplund functions on a topological group G (which contains the algebra WAP(G) of weakly almost periodic functions). We note that, in contrast to WAP(G) where the invariant mean is unique, for some groups (including the integers) there are uncountably many invariant means on Asp(G). Finally, we observe that dynamical systems in the larger class of tame G-systems need not admit an invariant probability measure, and the algebra Tame(G) is not left amenable.

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Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel

    Eli Glasner

  2. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Michael Megrelishvili

Authors
  1. Eli Glasner
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  2. Michael Megrelishvili
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Correspondence to Eli Glasner.

Additional information

This research was partially supported by Grant No 2006119 from the United States-Israel Binational Science Foundation (BSF).

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Glasner, E., Megrelishvili, M. On fixed point theorems and nonsensitivity. Isr. J. Math. 190, 289–305 (2012). https://doi.org/10.1007/s11856-011-0192-4

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  • DOI: https://doi.org/10.1007/s11856-011-0192-4

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