  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.
