Absolutely Singular Dynamical Foliations

By David Ruelle, Amie Wilkinson. 2000

94 citations recorded by the source corpus.

Abstract: Let A3 be the product of the automorphism of T2 and of the identity on T1. A small perturbation g of A3 among volume preserving diffeomorphisms will have an invariant family of smooth circles Γ forming a continuous foliation of T3. Corresponding to the vector bundle tangent to the circles Γ there is a “central” Lyapunov exponent of (g, volume), which is nonzero for an open set of ergodic g's. This surprising result of Shub and Wilkinson is complemented here by showing that the volume on T3 has atomic conditional measures on the Γ's: there is a finite k such that almost every Γ carries $k$ atoms of mass 1/k.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2da82b57127ce6674886720076d28bf332be7aa4

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Citations in the corpus

Averaging and Invariant Measures (2005, unknown): “Mostly contracting diffeomorphisms has been studied in [6], [9], [10], [11], [16], [42] and their properties are well understood.”
On differentiability of SRB states for partially hyperbolic systems (2004, background): “In particular, [54] gives an example of a perturbation where W c (f ε ) is absolutely singular for small ε = 0, that is, there exists a set Ω ε of total Lebesgue measure intersecting each leaf of W c (f ε ) by a set of zero leaf measure (see [53] for more discussion of this phenomenon).”
Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, background): “What is more, Ruelle and Wilkinson [49] showed that, in a similar setting, the disintegration of volume along center leaves is atomic, supported on finitely many points.”
Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics (2006, background): “Moreover, there exists a set E of full measure and an integer k > 1 such that E intersects almost every leaf W (x) at exactly k points (see [219]; the example in Section 10.”
Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins (2008, background): “Moreover, for a volume preserving perturbation of a skew product over the Anosov map of a torus, Shub– Wilkinson [10] and Ruelle–Wilkinson [8] discovered a set of full Lebesgue measure that intersects any central fiber in a finite set, whilst the union of all the central leaves constitutes the total”
Physical measures and absolute continuity for one-dimensional center direction (2013, methodology): “(See Pugh, Viana and Wilkinson [51].)”