Invariant measures for parabolic ifs with overlaps and random continued fractions

K. Simon, B. Solomyak, M. Urbanski

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36 Scopus citations

Abstract

We study parabolic iterated function systems (IPS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no *overlaps.* We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

Original languageEnglish
Pages (from-to)5145-5164
Number of pages20
JournalTransactions of the American Mathematical Society
Volume353
Issue number12
DOIs
StatePublished - 2001
Externally publishedYes

Keywords

  • Iterated function systems
  • Parabolic maps
  • Random continued fractions

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