Ergod. Th. & Dynam. Sys. (1982), 3,119-127 Printed in Great Britain Characteristic exponents of dynamical systems in metric spaces YURI KIFER Institute of Mathematics, Hebrew University, Jerusalem, Israel {Received 15 July 1982) Abstract. We introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are denned also for invariant sets. Section 1 Let 5' be a one-parameter continuous group of homeomorphisms of a metric separable space X with the distance d{-, •), where t is a continuous parameter -oo < t < oo or a discrete one r = ..., -1, 0,1, .... Assume that X has no isolated points then all sets B X (S, T) = {y eX\x:d(S'x,S'y)<8 for all re[O, T]} are non-empty. Set A s (x,t)= sup d(S'x,S'y)/d(x,y) (1) ysB x (5,r) and a s (x,t)= inf d{S'x,S'y)ld(x,y). (2) yeB x (S,[) Let p. be a Borel 5'-invariant probability measure such that f sup \\nA s {x,u)\n{dx)+ sup u f |ln A s {x, t)\^(dx)<oo. (3) This condition is satisfied, for instance, when there is K > 1 such that sup d(S u x,S u y)/d(x,y)^K (4) -lsusl for any x, y e X, x ^ y (some kind of Lipschitz condition on the transformations 5'). Indeed, from (4) it follows that for any x,yeX, x*y and all te (-00,00). Hence, |ln A s {x, r)|<(|/| + l) inK that gives (3). Define A«(x) = limsupj-rln A s (x, t) (5) f-*±oo \t\ Downloaded from https://www.cambridge.org/core. 13 Jun 2020 at 17:17:46, subject to the Cambridge Core terms of use.