Entropy of graphs, semigroups and groups

§0. Introduction Let X be a compact metric space and T : X → X is continuous transformation. Then the dynamics of T is a widely studied subject. In particular, h(T ) the entropy of T is a well understood object. Let Γ ⊂ X ×X be a closed set. Then Γ induces certain dynamics and entropy h(Γ). If X is a finite set then Γ can be naturally viewed as a directed graph. That is, if X = {1, ..., n} then Γ consists of all directed arcs i → j so that (i, j) ∈ Γ. Then Γ induces a subshift of finite type which is a widely studied subject. However, in the case that X is infinite, the subject of dynamic of Γ and its entropy are relatively new. The first paper treating the entropy of a graph is due to [Gro]. In that context X is a compact Riemannian manifold and Γ can be viewed as a Riemannian submanifold. (Actually, Γ can have singularities.) We treated this subject in [Fri1-3]. See Bullet [Bul1-2] for the dynamics of quadratic correspondences and [M-R] for iterated algebraic functions. The object of this paper is to study the entropy of a corresponding map induced by Γ. We now describe briefly the main results of the paper. Let X be a compact metric space and assume that Γ ⊂ X ×X is a closed set. Set Γ+ = {(xi)1 : (xi, xi+1) ∈ Γ, i = 1, ..., }. Let σ : Γ+ → Γ+ be the shift map. Denote by h(Γ) be the topological entropy of σ ∣Γ∞+ . It then follows that σ unifies in a natural way the notion of a (continuous) map T : X → X and a (finitely generated) semigroup or group of (continuous) transformations S : X → X. Indeed, let Ti : X → X, i = 1, ..., m, be m continuous transformations. Denote by Γ(Ti) the graphs corresponding to Ti, i = 1, ..., m. Set Γ = ∪1 Γ(Ti). Then the dynamics of σ is the dynamics of the semigroup generated by T = {T1, ..., Tm}. If T is a set of homeomorphisms and T −1 = T then the dynamics of σ is the dynamics of the group G(T ) generated by T . In particular, we let h(G(T )) = h(Γ) be the entropy of G(T ) using the particular set of generators T . For a finitely generated group G of homeomorphisms of X we define h(G) = inf T ,G=G(T ) h(G(T )).