Displaying similar documents to “The fractal dimension of invariant subsets for piecewise monotonic maps on the interval.”

Conformal measures for rational functions revisited

Manfred Denker, R. Mauldin, Z. Nitecki, Mariusz Urbański (1998)

Fundamenta Mathematicae

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We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.

Multifractal dimensions for invariant subsets of piecewise monotonic interval maps

Franz Hofbauer, Peter Raith, Thomas Steinberger (2003)

Fundamenta Mathematicae

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The multifractal generalizations of Hausdorff dimension and packing dimension are investigated for an invariant subset A of a piecewise monotonic map on the interval. Formulae for the multifractal dimension of an ergodic invariant measure, the essential multifractal dimension of A, and the multifractal Hausdorff dimension of A are derived.

Invariant measures for iterated function systems

Tomasz Szarek (2000)

Annales Polonici Mathematici

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A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.

Multifractal analysis for Birkhoff averages on Lalley-Gatzouras repellers

Henry W. J. Reeve (2011)

Fundamenta Mathematicae

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We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent.