Wild attractors and thermodynamic formalism
Abstract
Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family ƒλ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential φt = -t log |ƒλ'|, there is a unique phase transition at some t1 ≤ 1, and the pressure P(φt ) is analytic (with unique equilibrium state) elsewhere. The pressure is majorised by a non-analytic C∞ curve (with all derivatives equal to 0 at t1 < 1) at the emergence of a wild attractor, whereas the phase transition at t1 = 1 can be of any finite order for those λ for which ƒλ is Lebesgue conservative. We also obtain results on the existence of conformal measures and equilibrium states, as well as the hyperbolic dimension and the dimension of the basin of ω(c).
Citation
Bruin , H & Todd , M J 2015 , ' Wild attractors and thermodynamic formalism ' , Monatshefte für Mathematik , vol. 178 , no. 1 , pp. 39-83 . https://doi.org/10.1007/s00605-015-0747-2
Publication
Monatshefte für Mathematik
Status
Peer reviewed
ISSN
0026-9255Type
Journal article
Description
MT was partially supported by NSF Grants DMS 0606343 and DMS 0908093.Collections
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