## 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.

Original language | English |
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Pages (from-to) | 289-305 |

Number of pages | 17 |

Journal | Israel Journal of Mathematics |

Volume | 190 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2012 |

### Bibliographical note

Funding Information:∗This research was partially supported by Grant No States–Israel Binational Science Foundation (BSF). Received September 7, 2010

### Funding

∗This research was partially supported by Grant No States–Israel Binational Science Foundation (BSF). Received September 7, 2010

Funders | Funder number |
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United States-Israel Binational Science Foundation |