The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

Abstract

Abstract. We study the existence of solutions g to the functional inequality f ≤ gwhere f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an indifferent fixed point, and β is the maximum ergodic average of f . Using a method due to T. Bousch, we show that continuous solutions g always exist when the Hölder exponent of f is close to 1. In the converse direction, we construct explicit examples of continuous functions f with low Hölder exponent for which no continuous solution g exists. We give sharp estimates on the best possible Hölder regularity of a solution g given the Hölder regularity of f .

Ergodic optimization in dynamical systems

“…Morris [126] considered existence and non-existence of revelations in the context of circle maps with an indifferent fixed point (improving on earlier work [36,148]). Specifically, he considered expanding circle maps of Manneville-Pomeau type α ∈ (0, 1), generalising the Manneville-Pomeau map x → x + x 1+α (mod 1), and proved: The estimate on the Hölder exponent of the revelation in Theorem 6.7 is sharp: there exist γ-Hölder functions without any revelation of Hölder exponent strictly larger than γ − α (see [126]). Branco [35] has considered certain degree-2 circle maps with a super-attracting fixed point, proving that if f is α-Hölder, and the super-attracting fixed point is not maximizing, then there exists an α-Hölder revelation.…”

“…[126] (Revelation theorem: T of Manneville-Pomeau type) If T is an expanding circle map of Manneville-Pomeau type α ∈ (0, 1), then every Hölder function of exponent γ > α has a (γ − α)-Hölder revelation; however there exist α-Hölder functions which do not have a revelation.…”

Constrained ergodic optimization for asymptotically additive potentials

“…denote the set of all f −maximizing measures. The study of the variational problem of the functional β(•) and the set M max (•) has been termed ergodic optimization, and has attracted some recent research interest [6,7,8,10,11,16,20,21,22,23]. Analogous problems for sub-additive potentials has been investigated for deterministic dynamical systems in [17,30,31] and for random dynamical systems in [13].…”

Sturmian maximizing measures for the piecewise-linear cosine family

“…[29] for an overview). Most of this work has concerned theoretical aspects of the subject, including abstract information on the nature of maximizing measures [8,10,11,13,19,30,36,37,39,40,41,42,44,47,48,49,51], approximation of maximizing measures [9,15,18], connections with thermodynamic formalism [12,17,27,28,33,35,38,46], and connections with partial orders on M [2,31,32,34].…”