Probabilistic Potential Theory And Induction Of Dynamical Systems
Résumé
In this article, we outline a version of a balayage formula in probabilistic potential theory adapted to measure-preserving dynamical systems. This balayage identity generalizes the property that induced maps preserve the restriction of the original invariant measure. As an application, we prove in some cases the invariance under induction of the Green-Kubo formula, as well as the invariance of a new degree 3 invariant. The central objects of the probabilistic theory of potential [16, 4] are the solutions of the Poisson equation: (I − P)(f) = g, where P is the transition kernel of a Markov chain and g is fixed. Its solutions exhibit, in particular, an invariance under induction [16, Chapter 8.2]. Given a subset Ψ of the state space, if P Ψ is the transition kernel for the induced Markov chain, then one can deduce the solutions of the equation (I − P Ψ)f = g from those of the initial equation (I − P)(f) = g. This invariance, in turn, is a powerful tool to compute P Ψ , and from there hitting probabilities: if one is given a starting site and a number of targets, it is possible to compute the distribution of the first target hit by the Markov chain [17]. A number of physically or geometrically relevant dynamical systems, such as the Lorentz gas or the geodesic flow on abelian covers of hyperbolic manifolds, behave globally or locally like random walks. For instance, they satisfy global [12] and local central limit theorems, invariance principles [6], large deviations [18], etc. This raises the question of adapting the probabilistic potential theory to such systems. In a previous work [15], the authors devised a method related to this theory to estimate the hitting probability of a single far away target for such systems. It relied on a stronger form of invariance under induction satisfied by Green-Kubo's bilinear form: σ 2 GK (A, m, T ; f, f) := A f 2 d m +2 n≥1 A f · f • T n d m , (0.1) which appear is the limiting variance in the central limit theorem. While the method used in [15] does not extend to a larger number of targets, it suggests the possibility of applying potential theory to dynamical systems. In this article, we show how to adapt the invariance under induction of the Poisson equation to general recurrent measure-preserving dynamical systems. Let (A, m, T) be a measure-preserving recurrent dynamical system, with L the transfer operator associated to (A, m, T). Given B ⊂ A and T B the first return map of T to B, the system (B, m |B , T B) is recurrent; let L B be the associated transfer operator. Our first result shall be: Proposition 0.1. Let (A, m, T) be measure-preserving, with m a recurrent σ-finite measure. Let B ⊂ A be such that 0 < m(B) ≤ +∞. Let p ∈ [1, ∞] and f , g ∈ L p (A, m) be such that g ≡ 0 on B c and: (I − L)f = g. (0.2)
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