Ergodic Optimization for Hyperbolic Flows and Lorenz Attractors
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Annales Henri Poincar´ e
Ergodic Optimization for Hyperbolic Flows and Lorenz Attractors Marcus Morro, Roberto Sant’Anna and Paulo Varandas Abstract. In this article we consider the ergodic optimization for hyperbolic flows and Lorenz attractors with respect to both continuous and H¨ older continuous observables. In the context of hyperbolic flows we prove that a Baire generic subset of continuous observables have a unique maximizing measure, with full support and zero entropy, and that a Baire generic subset of H¨ older continuous observables admit a unique and periodic maximizing measure. These results rely on a relation between ergodic optimization for suspension semiflows and ergodic optimization for the Poincar´e map with respect to induced observables, which allow us to reduce the problem for the context of maps. Using that singular-hyperbolic attractors are approximated by hyperbolic sets, we obtain related results for geometric Lorenz attractors. Mathematics Subject Classification. Primary 37D20, 37C10, 37C50; Secondary 37C27, 37A05. Keywords. Ergodic optimization, Hyperbolic flows, Gluing orbit property, Lorenz attractors.
1. Introduction and Statement of the Main Results Let X be a compact metric space, f : X → X be a continuous map and Mf be the collection of f -invariant Borel probability measures on X. The objects of interest in the field of ergodic optimization are those f -invariant probability measures which maximize, or minimize, the space average ϕdμ, for ϕ : X → R, over all μ ∈ Mf . These are the maximizing measures, or minimizing, measures for the function ϕ (with respect to the dynamical system f ). As usual, we restrict our attention to maximizing measures, since a minimizing measure for ϕ is a maximizing measure for −ϕ. The compactness of Mf and continuity of the function μ → ϕdμ ensures that maximizing measures always exist. It is also clear from the ergodic decomposition theorem
M. Morro et al.
Ann. Henri Poincar´e
that almost all ergodic components of a maximizing measure are maximizing measures; hence, ergodic maximizing measures also exist. Hence, some of the fundamental question in ergodic optimization are: ◦ ◦ ◦ ◦
What can we say about the maximizing measures? Is there only one maximizing measure for typical observables? Can we describe the support of a maximizing measure? Are maximizing measures typically periodic?
There exists an extensive list of contributions to these problems built over different approaches, some of which inspired by statistical mechanics and thermodynamic formalism (zero temperature limits) and others from the theory of cohomology equations (construction of sub-actions). In the known situations, the answer to the previous questions usually depends on the class of the dynamics but also on the regularity of the observables. In [31] Ma˜ n´e conjectured that for a generic Lagrangian there exists a unique minimizing measure, and it is supported by a periodic orbit. Contreras, Lopes and Thieullen [17] and later Contreras [18] obtained a proof of this conjecture
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