Gibbs measures for partially hyperbolic attractors

Abstract

We consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.

Fast mixing for attractors with a mostly contracting central direction

“…As a by-product of our construction (see §2.3), we also obtain that there exists a set A ⊂ such that: • ∞ j =0 f j (A) contains except for a zero set (with respect to the SRB measure or any uu-Gibbs measure-see [12]); • any point x ∈ A presents a central stable disk γ (x) x such that the radius of γ (x) is greater than r 0 and As a by-product of our construction (see §2.3), we also obtain that there exists a set A ⊂ such that: • ∞ j =0 f j (A) contains except for a zero set (with respect to the SRB measure or any uu-Gibbs measure-see [12]); • any point x ∈ A presents a central stable disk γ (x) x such that the radius of γ (x) is greater than r 0 and…”

A Mechanical Model for Fourier’s Law of Heat Conduction

“…An ( f t )-invariant probability measure ρ on S is called (by Pesin and Sinai [20]) a u-Gibbs state if the conditional measures on the local unstable manifolds have a density of a certain canonical form. The u-Gibbs states are precisely the ( f t )-invariant probability measures ρ on S which are absolutely continuous with respect to the foliation W u (i.e., if X ∩ W x has leaf Lebesgue measure 0 for each local unstable manifold W x , then ρ(X ) = 0).…”

“…The perturbed time evolution ( f t ) is thus also partially hyperbolic, provided λ is small enough (see [20]). For λ = 0 there need not be an SRB state, but limits of T are u-Gibbs states, and the following applies:…”

Non-Gibbsianness of SRB measures for the natural extension of intermittent systems

“…As we will see in §3, for both the complex continued fraction and the inhomogeneous Diophantine algorithm, rates of decay of σ (n) and σ − (n) are polynomial. It follows from these observations that the natural extension µ of µ gives an analogy of u-Gibbs measures for partially hyperbolic systems introduced by Pesin and Sinai in [14]. In [21], a version of local product structure (the so-called weak local product structure) was established for the invertible extension of invariant ergodic weak Gibbs measures µ for φ of WBV with P top (T , φ) < ∞.…”

“…In the category of smooth dynamical systems with non-zero Lyapunov exponent, u-Gibbs measures introduced by Pesin and Sinai in [14] played important roles in establishing ergodic and further statistical properties [1,3,4]. Here, u-Gibbs measures are invariant probability measures whose conditional measures along corresponding Pesin unstable leaves are absolutely continuous with respect to Lebesgue measure on the leaves.…”