Title:

On the ergodic theory of cellular automata and twodimensional Markov shifts generated by them

In this thesis we study measurable and topological dynamics of certain classes of cellular automata and multidimensional subshifts. In Chapter 1 we consider onedimensional cellular automata, i.e. the maps T: PZ > PZ (P is a finite set with more than one element) which are given by (Tx)i==F(xi+1, ..., xi+r), x=(xi)iEZ E PZ for some integers 1≤r and a mapping F: Pr1+1 > P. We prove that if F is right (left) permutative (in Hedlund's terminology) and 0≤10 and T is surjective, then the natural extension of the system (PZ, B, μ, T) is a Kautomorphism. We also prove that the shift Z2action on a twodimensional subshift of finite type canonically associated with the cellular automaton T is mixing, if F is both right and left permutative. Some more results about ergodic properties of surjective cellular automata are obtained Let X be a closed translationally invariant subset of the ddimensional full shift PZd, where P is a finite set, and suppose that the Zdaction on X by translations has positive topological entropy. Let G be a finitely generated group of polynomial growth. In Chapter 2 we prove that if growth(G) = = ⊥c (Zp)Z2 defined by the principal ideals c Zp [u±1, v±t] ≃ ((Zp)Z2)^ with f(u, v) = cf(0,0) + cf(1,0)u + cf(0,1)v, cf(i, j) E Zp\{0}, on which Z2 acts by shifts. We give the complete topological classification of these subshifts with respect to measurable isomorphism.
