On fixed point theorems and nonsensitivity

Details On fixed point theorems and nonsensitivity On fixed point theorems and nonsensitivity

2010-07-29

arxiv.org/abs/1007.5303v2

Mathematics

Dynamical Systems

12 pages, revised version, to appear in Israel J. of Math

Scientific paper

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), for some groups 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.

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