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A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. (English) Zbl 1255.47053

Let \(T\) be a non-expansive map of a complete metric space \((X,d)\) into itself and consider the linear escape rate of the orbits of \(T\): \(\rho(T)= \lim_{k\to+\infty} {d(x,T^kx)\over k}\). The main result of the paper is the following maximum characterization of \(\rho(T)\). If \((X,d)\) is metrically star-shaped, then \[ \inf d(y, Ty)= \rho(T)= \max_h\underset{x\in X}\inf(h(T(x))- h(x)), \] where the maximum is attained over the set of Martin functions \(h\) of \((X,d)\). Moreover, if \(\rho(T)> 0\), then any function attaining the maximum is a horofunction. An analogous result is stated for hemi-metric spaces.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54E40 Special maps on metric spaces
54H25 Fixed-point and coincidence theorems (topological aspects)
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