Strongly mixingg-measures

Abstract

LetT be ann-to-1 covering transformation of the compact metric spaceX (e.g. (X, T) then-shift). For suitable functionsg onX an “inverse”ϕ _g ofT is defined:ϕ _g is a Markov kernel. Ifg is strictly positive and satisfies a Lipschitz condition, then there exists a uniqueϕ _g measure, strongly mixing underT. Conversely, we associate to anyT-invariant probability measure a suitableg, and ifg is “nice”, then strong mixing is present. Examples include all Bernoulli and Markov measures on then-shift. The strong mixing criterion is useful, and applications to harmonic analysis, ergodic theory, and symbolic dynamics are given. For example: if$\mathfrak{G}$ is any infinite subgroup of the group of roots of unity, there exist uncountably many (explicitly constructible) continuous Morse sequences whose corresponding dynamical systems are pairwise non-isomorphic and all have as eigenvalue group exactly the given group$\mathfrak{G}$.