On Rigidity of Algebraic Dynamical Systems

We study topological rigidity of algebraic dynamical systems. In the first part of this paper we give an algebraic condition for rigidity that unifies previous rigidity results and we settle an old question of Walters. In the second part we consider the rigidity of hyperbolic systems. An algebraic dynamical system (X, S) has a phase space X that is a compact abelian group and a transformation semigroup S acts on X by affine transformations. In this paper X is always connected and metrisable, S is a discrete commutative semigroup with unity (a monoid) and the action of S preserves the Haar measure. A continuous map f : X → Y between phase spaces of S-algebraic dynamical systems is equivariant if f(s · x) = s · f(x) for all s ∈ S. An algebraic dynamical system (Y, S) is topologically rigid if all equivariant maps into Y are affine; i.e., if f : X → Y is equivariant for any X, then f(x) = y +A(x) for some y ∈ Y and a homomorphism A between X and Y . Usually, the notion is restricted to a subclass of all S-algebraic dynamical systems. For instance, if (Y, S) is ergodic, then all (X,S) are assumed to be ergodic as well. Let Z[S] be the semigroup ring that consists of all formal sums of elements of S with the natural multiplication. If S acts by endomorphisms then X has the structure of a Z[S]module. An algebraic dynamical system does not necessarily have this structure since S acts by affine transformations, but this can be adjusted. If T is affine then T (x) = a+A(x) for some homomorphism A and we say that A is the linear part of T . If S acts on X by affine transformations, then we can give X the structure of a Z[S]-module by replacing the affine transformations x 7→ s · x by their linear parts. By Pontryagin duality this induces a Z[S]-module structure on the character group X̂. The linear space C(X,R) of real-valued continuous maps inherits an S-action that is defined by f(x) 7→ f(s · x). These are endomorphisms, hence C(X,R) is a Z[S]-module. The submodule C0(X,R) consists of all functions such that ∫ fdμ = 0 for the normalized Haar measure μ. In the first part of the paper we show that the rigidity of Y depends on the Z[S]-module structure of the C0(X,R), where X ranges over the subclass of S-dynamical systems. More specifically, we obtain the following algebraic characterization of rigidity that unifies previous results of [1, 7, 2, 3]. Theorem 1. Let μ be the Haar measure on X and let C0(X,R) ⊂ C(X,R) be the submodule of all functions for which ∫ fdμ = 0. All equivariant maps from X to Y are affine if and only if the only Z[S]-module homomorphism between Ŷ and C0(X,R) is the trivial homomorphism. We are indebted to [3], which implicitly contains this theorem for the case that S is equal to Zd and acts by endomorphisms. The techniques we use can be viewed as an extension of the techniques used in [2]. 1991 Mathematics Subject Classification. Primary 54 H15; Secondary 37 B10, 11 R04.