Some combinatorial results in topological dynamics

Abstract

The main results of this thesis are combinatorial in nature. We will be mainly working with the continuous automorphisms on the torus T2 and with the continuous self maps on R. The thesis is conveniently divided into _ve chapters. The general mathematical setting is that of an abstract dynamical system with discrete time parameter, that is, a pair (X; f) where X is a topological space and f a continuous mapping of X into itself. We are interested in the action of the iterates of f on X.Chapter-1 is introductory in nature. We explain the basic notions of discrete dynamical systems and some important results emphasizing the role of the set of periodic points and the set of periods in chaos. We discuss briey about the de_nitions of chaos due to Devaney and Li-Yorke. It is already known [21] that for a hyperbolic (having no eigen values on the unit circle) continuous toral automorphism, the periodic points are precisely the rational points. In chapter-2, we calculate the set of periodic points for other continuous toral automorphisms; this happens to be the subgroup generated by Q _ Q[ (a line with rational slope). In fact, for all non-hyperbolic continuous toral automorphisms, there are uncountably many periodic points. Also, we prove that: every such subgroup is the set of periodic points for some continuous toral automorphism. In Chapter-3, we _rst discuss some well known results about Per(f) due to Sharkovski [17], Baker [5] and many others; our main result of this chapter is similar in spirit to these. We prove that there are exactly 8 subsets of N which can occur as Per(T) for some continuous toral automorphism T. We solve the problem separately for hyperbolic and nonhyperbolic automorphisms. It is interesting that, for nonhyperbolic toral automorphisms, we are able to list the set Per(T) in terms of the minimal polynomial of T. In chapter-4, we introduce the notion of special points and nonordinary points of a dynamical system. These notions are new to the literature, though they arise very naturally.