In this paper we investigate the growth of finitely generated groups. We recall the definition of the algebraic entropy of a group and show that if the group is acting as a discrete subgroup of the isometry group of a Cartan–Hadamard manifold with pinched negative curvature then a Tits alternative is true. More precisely the group is either virtually nilpotent or has a uniform growth bounded below by an explicit constant.
We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the -measure.
We prove that by considering a finitary (almost continuous) symbolic extension of a topological dynamical system instead of a continuous extension, one cannot achieve any drop of the entropy of the extension.