Almost automorphic and bijective factors of substitution shifts
Reem Yassawi (QMUL)
Let f:(X,T)—>(Y,S) 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. We say that (X,T) is an isometric extension of (Y,S) if there is a continuous TxT-invariant real-valued map R on {(x,x’): f(x)=f(x’)} which is a metric when restricted to f-fibres.
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. We investigate this result for the special family of constant length substitution shifts. Our approach is algebraic: we characterise the existence of factors (Y, S) of (X,T), such that (Y,S) is an almost automorphic extension of the maximal equicontinuous factor of (X,T), in terms of a finite semigroup defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green’s R-relation and the existence of bijective factors, which can lead to isometric extensions, in terms of Green’s L-relation. Our results are constructive. This is joint work with Álvaro Bustos-Gajardo and Johannes Kellendonk.
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