Dynamical Systems seminar - Almost automorphic and bijective factors of substitution shifts

Speaker: Reem Yassawi (Queen Mary University of London)
 
Abstract: Let f:(X,T)—>(Y,T') be a factor map of topological dynamical systems. We say that  (X,T) is an  almost automorphic extension if for some y in Y, the f-preimage of y is a singleton.  In the case where  (X,T) is  a finite-to-one extension of  (Y,T'), we say that this extension is  isometric   if f is k-to-1 for some k. Finally, a point x in X is distal if  there is no point whose T-orbit comes arbitrarily close to that of x. Veech’s theorem tells us that any system with a residual set of distal points has an almost automorphic extension  which can be realised as an inverse limit of alternating isometric and almost automorphic extensions of the trivial (one point) system. We investigate this result for the special family of constant length substitution shifts. Our approach is algebraic: we define a finite semigroup S defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green’s R-relation of S, and the existence of factors, which can lead to isometric extensions, in terms of Green’s L-relation of S. Our results are constructive. This is joint work with Álvaro Bustos-Gajardo and Johannes Kellendonk. I promise my talk will be elementary!