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).