The Hausdorff Dimension of an Ergodic Invariant Measure for a Piecewise Monotonic Map of the Interval

Abstract

We consider a piecewise monotonie and piecewise continuous map T on the interval. If T has a derivative of bounded variation, we show for an ergodic invariant measure μ with positive Ljapunov exponent λμ that the Hausdorff dimension of μ equals hμ / λμ.

Multifractal dimensions for invariant subsets of piecewise monotonic interval maps

“…If e g j is of bounded p-variation for some p > 0, then the claim follows from Lemma 1 of [4]. Otherwise …”

“…for every x ∈ C. A proof analogous to the first part of the proof of Proposition 2 of [4] shows the existence of a set L ⊆ [0, 1] with µ(L) = 1 such that for every x ∈ L there exists a strictly increasing sequence (n k (x)) k∈N in N with…”

“…For s = 0 this formula is well known (cf. [4]). Similar results can also be found in [7] (even for higher dimensional systems), but only in situations where the transformation admits a Markov partition.…”

“…in the Markovian case the "bounded distortion principle" is frequently used) we use an approximation by Markov maps and the methods developed in [4].…”

On cusps and flat tops

“…With unstable manifold in hand, we use regularly returning (or nice) intervals to give simple proofs of the dynamical volume lemma in Proposition 6.2 and of the existence of a Pesin partition in Proposition 7.1, compare [17] and [21].…”