Skip to main content
Log in

Strongly mixingg-measures

  • Published:
Inventiones mathematicae Aims and scope


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\(\mathfrak{G}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Ionescu-Tulcea, C. T.: On a class of operators occurring in the theory of chains of infinite order. Canad. J. Math.11, 112–121 (1969).

    Google Scholar 

  2. Kakutani, S.: Ergodic theory of shift transformations. Proc. V. Berk. Sym.11, 405–414 (1967).

    Google Scholar 

  3. Karlin, S.: Some random walks arising in learning models I. Pac. J. Math.3, 725–756 (1953).

    Google Scholar 

  4. Keane, M.: Generalized Morse sequences. Z. Wahrscheinlichkeistheorie verw. Geb.10, 335–353 (1968).

    Google Scholar 

  5. Norman, M. F.: Some convergence theorems for stochastic learning models with distance diminishing operators. Journal of Math. Psych.5, 61–101 (1968).

    Google Scholar 

  6. Norman, M. F.: A uniform ergodic theorem for certain Markov operators on Lipschitz functions on bounded metric spaces. Z. Wahrscheinlichkeitstheorie verw. Geb.15, 51–66 (1970).

    Google Scholar 

  7. Mandrekar, V., Nadkarni, M.: On ergodic quasi-invariant measures on the circle group. J. Funct. Anal.3, 157–163 (1969).

    Google Scholar 

  8. Zygmund, A.: Trigonometric series I, 208ff. Cambridge: University Press 1968.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

C.N.R.S. Équipe Associée 250.

Research supported in part by NSF grant GP-16392 while the author was visiting at Yale University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Keane, M. Strongly mixingg-measures. Invent Math 16, 309–324 (1972).

Download citation

  • Received:

  • Issue Date:

  • DOI: