Mather measures and the Bowen–Series transformation
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.