TY - JOUR

T1 - On fixed point theorems and nonsensitivity

AU - Glasner, Eli

AU - Megrelishvili, Michael

N1 - Funding Information: ∗This research was partially supported by Grant No States–Israel Binational Science Foundation (BSF). Received September 7, 2010

PY - 2012/8

Y1 - 2012/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84865566499&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s11856-011-0192-4

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

M3 - مقالة

SN - 0021-2172

VL - 190

SP - 289

EP - 305

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -