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.
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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.
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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.
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Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and $h_{top}(T) > 0$, it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.
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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.
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