The thermodynamic formalism of random walks: Relevance for chaotic diffusion and multifractal measures

We apply the thermodynamic formalism to discrete random walks on infinite lattices. This can be considered as a new application of the thermodynamic formalism, but it also provides a very general framework for the thermodynamics of chaotic systems and multifractals. In contrast to finite Markov models which are often discussed in this context, nonanalytic behavior of characteristic quantities is encountered for models with broken translation invariance. In particular, we present new mechanisms for second-order phase transitions in the Lyapunov fluctuation spectra for chaotic diffusion and the singularity spectra of multifractal measure.