In this paper we introduce topological and algebraic structures on the space of DNA sequences. The idea of these structures comes from genetic recombination.

We introduce and study the topological concepts of ergodic shadowing, chain transitivity and topological ergodicity for dynamical systems on non-compact non-metrizable spaces. These notions generalize the relevant concepts for metric spaces. We prove that a dynamical system with topological ergodic shadowing property is topologically chain transitive, and that topological chain transitivity together with topological shadowing property implies topological ergodicity.

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Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

Abstract. In this paper we present invariant dynamical properties under G-conjugacy. Moreover we introduce the notions of G-minimal systems, strong G-shadowing property and limit G-shadowing property and we show that these properties are invariant under topological Gconjugacy.