Currently displaying 1 – 17 of 17

Showing per page

Order by Relevance | Title | Year of publication

Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II

Franz Hofbauer — 1993

Studia Mathematica

We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.

The recurrence dimension for piecewise monotonic maps of the interval

Franz Hofbauer — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [ 0 , 1 ] , giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure...

Multifractal spectra of Birkhoff averages for a piecewise monotone interval map

Franz Hofbauer — 2010

Fundamenta Mathematicae

We study the entropy spectrum of Birkhoff averages and the dimension spectrum of Lyapunov exponents for piecewise monotone transformations on the interval. In general, these transformations do not have finite Markov partitions and do not satisfy the specification property. We characterize these multifractal spectra in terms of the Legendre transform of a suitably defined pressure function.

Hausdorff and packing dimensions for ergodic invariant measures of two-dimensional Lorenz transformations

Franz Hofbauer — 2009

Commentationes Mathematicae Universitatis Carolinae

We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence...

Multifractal dimensions for invariant subsets of piecewise monotonic interval maps

Franz HofbauerPeter RaithThomas Steinberger — 2003

Fundamenta Mathematicae

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.

Page 1

Download Results (CSV)