Lifts of strictly ergodic subshifts by permutative sliding block codes

Abstract

Permutative sliding block codes - in the sense of Hedlund - give rise to finite-to- one extensions of subshifts. We provide criteria under which minimality and unique ergodicity are preserved by this 'lifting procedure'. These findings are illustrated by means of some nat- ural example families, which we use to obtain strictly ergodic finite-to-one extensions of sub- stitution subshifts (including the case of Fibonacci, Tribonacci, silver mean and noble means substitutions). We further study the interplay between permutative sliding block codes and Toeplitz flows. In particular, we find examples which demonstrate that a permutative lift of a strictly ergodic subshift may be minimal, but not uniquely ergodic - a phenomenon which cannot occur for primitive substitution subshifts.