Generic Properties of Invariant Measures for Continous Piecewise Monotonic Transformations.
ß-Shifts Have Unique Maximal Measure.
The Topological Entropy of the Transformation x ... ax (1-x).
The Box Dimension of Completely Invariant Subsets for Expanding Piecewise Monotonic Transformations.
Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval
Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.
Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II
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.
Monotonic mod one transformations
The recurrence dimension for piecewise monotonic maps of the interval
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 , 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
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.
Erratum to "Multifractal spectra of Birkhoff averages for a piecewise monotone interval map" (Fund. Math. 208 (2010), 95-121)
Hausdorff and packing dimensions for ergodic invariant measures of two-dimensional Lorenz transformations
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...
Topologically Transitive Subsets of Piecewise Monotonic Maps, Which Contain no Periodic Points.
Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations.
Zeta-functions and transfer-operators for piecewise linear transformations.
Density of periodic orbit measures for transformations on the interval with two monotonic pieces
Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and , it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.
Some remarks on recent results about S-unimodal maps
Multifractal dimensions for invariant subsets of piecewise monotonic interval maps
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.
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