In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is c 0 or ℓ p (1 ≤ p < ∞), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.

The notion of Haar null set was introduced by J. P. R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke. During the last twenty years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. Inspired by these various results, we introduce the topological analogue of the notion of Haar null set. We call it Haar meager set. We prove some basic properties of this notion, state some open problems and suggest a possible line of investigation which may lead to the unification of these two notions in certain context.2010 Mathematics Subject Classification. 54E52, 54H11, 28C99.

In this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of spaces H({0, 1} N ) and C({0, 1} N ), the space of homeomorphisms and the space of self-maps of the Cantor space, respectively. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the comeager conjugacy class of the space H({0, 1} N ). The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is a comeager subset of H({0, 1} N ) each element of which has topological entropy zero. Our analogous investigation in C({0, 1} N ) yields a surprising result: there is a comeager subset of C({0, 1} N ) such that any two elements of this set are conjugate to each other by an element of H({0, 1} N ). Our description of this class also yields many old and new results concerning dynamics of a comeager subset of C({0, 1} N ).2000 Mathematics Subject Classification. Primary: 37B99, 54H20, Secondary: 22D05, 05C20.

Abstract. We study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics. We provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes. For dendrites with finitely many branch points the homeomorphism and quasi-homeomorphism classes coincide, and the minimal quasihomeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the homeomorphism relation between dendrites is S ∞ -universal. It is shown that the classes of trees and graphs are both D 2 (Σ 0 3 )-complete, the class of dendrites is Π 0 3 -complete, and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is Π 0 3 -complete. We also show that if G is a nondegenerate finitely triangulable continuum, then the class of G-like continua is Π 0 2 -complete.