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Title: On the ergodic theory of cellular automata and two-dimensional Markov shifts generated by them
Author: Shereshevsky, Mark Alexandrovich
ISNI:       0000 0001 3404 1423
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1992
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In this thesis we study measurable and topological dynamics of certain classes of cellular automata and multi-dimensional subshifts. In Chapter 1 we consider one-dimensional 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: Pr-1+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 Z2-action on a two-dimensional 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 d-dimensional full shift PZd, where P is a finite set, and suppose that the Zd-action 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.
Supervisor: Not available Sponsor: University of Warwick
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics