Extremal Lyapunov exponents: an invariance principle and applications

By Artur Avila, Marcelo Viana. 2010

187 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d6f22714231f89867964b3754880bf35654c9a9f

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In Mathematical Physics

1. Artur Avila
2. Marcelo Viana
3. Jiagang Yang
4. Carlos Matheus
5. Martin Möller
6. A. Eskin
7. G. Forni
8. F. R. Hertz

In Statistical and Nonlinear Physics

1. Artur Avila
2. Jiagang Yang
3. Marcelo Viana
4. F. R. Hertz
5. Amie Wilkinson
6. Carlos Matheus
7. Martin Möller
8. R. Ures

Citations in the corpus

• Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards (2013, background): “Viana [9] for some illustrations of this phenomenon).”
• Physical measures and absolute continuity for one-dimensional center direction (2013, methodology): “On the other hand, we felt that recent progress in the theory of cocycles, specially [13,14] could shed some light into such situations.”
• Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle (2012, methodology): “he projective action of the groupoid generated by the holonomies does not preserve some measure, a property which is commonly used to establish the non-triviality of Lyapunov spectra, see e.g. [BGV], [AV] and references therein. Acknowledgements. The authors would like to thank Alex Wright for makin”
• Continuity of Lyapunov Exponents for Cocycles with Invariant Holonomies (2015, background): “same corollary for DfjEs instead, provided that dimEs= 2. Remark 3.5. The conclusions of Corollary3.4can be extended to 2-dimensional cocycles over maps f: X!Xwhich are hyperbolic homeomorphisms (see [AV10]) with Xa compact metric space. This includes the derivative cocycle of a dieomorphism f : M !”
• Complex one-frequency cocycles (2013, background): “ conservative diffeomorphisms for which the Oseledets splitting is not dominated. However, as far as we know, all such examples currently rely on some underlying uniform form hyperbolicity (see, e.g., [AV], [ASV]). It would seem that similar considerations apply to the case of cocycles over hyperboli”
• Continuity of Lyapunov exponents for random two-dimensional matrices (2016, background): “Moreover, this has been extended to cocycles and diffeomorphisms in arbitrary dimension, by Bochi and Viana [6, 7].”
• Measure rigidity for random dynamics on surfaces and related skew products (2015, background): “d ˆν-a.s. invariant. Indeed, if all exponents are positive, then the measures µξ are finitely supported for a.e. ξ. That µˆ is νˆ-a.s. invariant follows, for instance, from the invariance principle in [AV], the Fˆ-measurability of the measureµξ, and an argument similar to Proposition 4.6 above. 4.4. ”
• Synchronization (2018, unknown): “For a real parameter P 0 > 0 , define Then (1.1) holds and we have”
• Maximizing measures for partially hyperbolic systems with compact center leaves (2010, methodology): “To prove the above claim, we first apply the invariance principle of Avila–Viana [1] to conclude that μ admits a disintegration [x→ μx ] which is s-invariant and u-invariant, and x→ μx varies continuously with x on supp(π∗(μ))= M.”
• Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, methodology): “Also important for our proofs here are the results of Avila, Santamaria and Viana [3] for cocycles over volume-preserving partially hyperbolic systems, both linear and smooth.”