Stationary Measures for Projective Transformations: The Blackwell and Furstenberg Measures

B. Bárány, M. Pollicott, K. Simon

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

In this paper we study the Blackwell and Furstenberg measures, which play an important role in information theory and the study of Lyapunov exponents. For the Blackwell measure we determine parameter domains of singularity and give upper bounds for the Hausdorff dimension. For the Furstenberg measure, we establish absolute continuity for some parameter values. Our method is to analyze linear fractional iterated function schemes which are contracting on average, have no separation properties (that is, we do not assume that the open set condition holds, see Hutchinson in Indiana Univ. Math. J. 30:713-747, 1981) and, in the case of the Blackwell measure, have place dependent probabilities. In such a general setting, even an effective upper bound on the dimension of the measure is difficult to achieve.

Original languageEnglish
Pages (from-to)393-421
Number of pages29
JournalJournal of Statistical Physics
Volume148
Issue number3
DOIs
Publication statusPublished - Aug 1 2012

Keywords

  • Blackwell measures
  • Entropy
  • Furstenberg measures
  • Hidden Markov chain
  • Singularity and absolute continuity of measures

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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