Pathological foliations and removable zero exponents

By M. Shub, Amie Wilkinson. 2000

184 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/1961439afb71a67989d4b306ba7cbfd7a8c16771

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

Citations in the corpus

Diffeomorphisms with positive metric entropy (2014, methodology): “arise. While it is understood that the \opposite" behavior, the vanishing of all Lyapunov exponents, does appear robustly (through the KAM mechanism), it has been proposed by Shub and Wilkinson ([SW], Question 1a) that for typical orbits of a generic Cr conservative dynamical system, the presen”
Physical measures and absolute continuity for one-dimensional center direction (2013, background): “(See Pugh, Viana and Wilkinson [51].)”
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).”
New criteria for ergodicity and nonuniform hyperbolicity (2009, methodology): “Shub and Wilkinson [34] have shown that j can be taken in such a way that the average center Lyapunov exponent is greater than zero.”
Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics (2006, background): “The proof in the general case is a modification of the argument in [229] (see also [23] and [74]).”
On mixing properties of compact group extensions of hyperbolic systems (2002, background): “Must a stably ergodic diffeomorphism be Bernoulli? stably Bernoulli? At present not much is known about stably Bernoulli systems apart from some examples constructed in [2, 5, 45].”
On the ergodicity of partially hyperbolic systems (2005, background): “[ SW2 ]. Brin and Pesin’s theorem applies in particular to the time-1 map ’1 of the geodesic o w for a compact surface of constant negative curvature.”
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”
On the uniform hyperbolicity of some nonuniformly hyperbolic systems (2001, background): “Shub and Wilkinson [13] gave examples of diffeomorphisms with nonzero Liapunov exponents along the ECu direction on Lebesgue almost every point which are not uniformly hyperbolic.”
A survey on partially hyperbolic dynamics (2006, background): “to make this perturbation is along the lines of [ ShWi2 ].”
Averaging and Invariant Measures (2005, unknown): “Similar computations in slightly different setting can be found in [43], [41], [19], [21].”
Every compact manifold carries a completely hyperbolic diffeomorphism (2002, background): “If f admits a dominated splitting then so does any dieomorphism which is suciently close to f. Shub and Wilkinson [ SW ] has shown that if TK = E1 E2 E3 where E1 SW ].”
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.”
Absolutely Singular Dynamical Foliations (2000, background): “= R2/Z2, given by 2 1 1 1 ! . Let A3 be the automorphism of the 3-torus T3 = R3/Z3 given by A2 0 0 1 ! . Let Diﬀ2 µ(T 3) be the set of C2 diﬀeomorphisms of T3 that preserve LebesgueHaar measure µ. In [SW1], M. Shub and A. Wilkinson prove the following theorem. Theorem: Arbitrarily close to A3 there ”
What Are SRB Measures, and Which Dynamical Systems Have Them? (2002, background): “I mention in particular [PS2], [ SW ], [D2] and [R4], which contain promising ideas.”