In this talk, I will report some recent progresses on multifractal analysis, thermodynamic formalism and refined large derivation (Chernov bound) on Holder continuous potentials for a class of branched covering map on topological 2-sphere called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumptions, initially investigated by M. Bonk and D. Meyer. The expanding Thurston maps we consider includes those that are not topological conjugate to rational maps, in particular they can have periodic critical points. As an application, we show a precise asymptotic for law of large number. This is a joint work with Zhiqiang Li and Xianghui Shi. 

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