On Expansive Transformation Groups
Ping-Fun Lam · Transactions of the American Mathematical Society · 1970
The shift of a symbolic flow is an expansive homeomorphism.A typical example of an expansive homeomorphism acting on a manifold is any member of a unimodular group without eigenvalues of absolute value 1 acting in the usual way on a «-dimensional torus («2:2).The latter is also an expansive automorphism acting on a compact group.The shift of a /j-adic solenoid is another such automorphism.In this paper we obtain some partial results on the following two problems: (A) to characterize and to classify all uniform spaces which admit an expansive homeomorphism or a positively expansive map, (B) to characterize and,to classify all topological groups which admit an expansive automorphism or an expansive endomorphism.Among other results we give an affirmative answer to the following better known open question.Question.If a compact connected group admits an expansive automorphism, must it be abelian ?We obtain, in fact, a much more general result (cf.Theorem 3.2).The question was raised in [4], where the case for compact connected Lie groups was also answered affirmatively.The case for finite-dimensional compact connected groups was proved by Wu [17].Using Wu's result, Eisenberg [6] shows also that if a compact connected finite-dimensional group admits a surjective expansive endomorphism, then it is abelian.Theorem 3.2 includes all these results.In Theorem 3.5 we slightly generalize Theorem 3.2 from compact groups to maximally almost periodic groups.The other results that we have are, for most part, improvement of the previously known results on (A).2. Expansive family of continuous maps on uniform spaces.2.1.Definition.Let X be a uniform space with uniformity <^.A nonempty family of continuous maps, 3F, of X into.X is said to be expansive, if there exists uEÎf such that for every x,y e X, x^y, there is/eJ^ depending on x and y so that (f(x), f(y)) <£ a.The uniform index a, which is not unique, is then called an expansive index of &