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 .