Generalized Farey trees, transfer Operators and phase transitions

Details Generalized Farey trees, transfer Operators and phase transitions Generalized Farey trees, transfer Operators and phase transitions

2006-06-06

arxiv.org/abs/math-ph/0606020v1

Physics

Mathematical Physics

39 pages, 10 figures

Scientific paper

10.1007/s00193-007-0090-1

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter map is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral analysis of generalized transfer operators. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.

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