Abstract. We prove that, for every extension of Banach algebras 0 → B → A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology groups. It follows from the continuous version of a result of Wodzicki that associated long exact sequences exist. In particular, they exist for every extension of C ∗-algebras. 1.

M.Sc. program is a joint program with Carleton University, administered by the Ottawa-

The (ǫ, n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance ǫ during the time interval n. Behavior of the (ǫ, n)-complexity functions as n → ∞ is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of (ǫ, n)-separated sets. We study such measures. In particular, we prove that they are invariant if the (ǫ, n)-complexity function grows subexponentially.

Abstract. In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closed-open subsets of X. Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact with respect to the compactopen topology. In case G is locally compact we decompose the space X into closed-open invariant disjoint sets that are related to various limit behaviors of the orbits. We also provide a simple example of a locally compact separable and complete metric space X with discrete group of isometries G such that the natural action of G on X has closed and non-closed orbits. 1.