Cohomological properties of Borel automorphisms and substitutions on infinite alphabet

This thesis is divided into two parts. In the first part we study cohomology of countable groups of Borel automorphisms of a standard Borel space. We prove that for a hyperfinite group of Borel automorphisms, the subgroup of coboundaries is dense in the group of cocycles. We study the cohomology of the 2-odometer from Borel perspective and describe all cocycles of 2-odometer taking values in a locally compact second countable group G. We show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup H of G. We study cocycles of a minimal homeomorphism of Polish space and provide a Borel version of Gottschalk-Hedlund theorem in this context. We study sets of real valued coboundries of two commuting Borel automorphisms. We show that if S,T are two commuting automorpisms of standard Borel space, such that they generate a free Borel Z^2-action then S and T do not have same sets of real valued coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel Z^d-actions. In the second part we study substitution dynamical systems on infinite alphabets from the perspective of Borel dynamics. We prove two versions of Rokhlin’s lemma for subshifts associated with substitutions on infinite alphabets. We construct stationary (and non-stationary) Bratteli-Vershik models for such substitution dynamical systems. Using the Bratteli-Vershik model we give an explicit formula for a shift-invariant measure (finite and infinite) and provide a criterion for this measure to be ergodic (or uniquely ergodic).