Renormalisation-Induced Phase Transitions for Unimodal Maps

Abstract

The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential −t log | Df |. Multiple and indeed infinitely many phase transitions at positive t can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.

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Correspondence to Neil Dobbs.

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The author was supported by the EU training network “Conformal Structures and Dynamics”.

Communicated by G. Gallavotti

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Dobbs, N. Renormalisation-Induced Phase Transitions for Unimodal Maps. Commun. Math. Phys. 286, 377–387 (2009). https://doi.org/10.1007/s00220-008-0656-5

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Keywords

  • Lyapunov Exponent
  • Topological Entropy
  • Pressure Function
  • Periodic Attractor
  • Markov Shift