Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding interval maps with finitely many branches

In this paper, we perform a multifractal analysis of Birkhoff averages for interval maps with finitely many branches and parabolic fixed points. Using the thermodynamic approach, we strengthen the results of Johansson et al. on the conditional variational principle for the multifractal spectra of Birkhoff averages. To do this, we develop several refined properties of the thermodynamic formalism for non-uniformly expanding interval maps.