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Invariance principle and non-compact center foliations (2210.14989v2)

Published 26 Oct 2022 in math.DS

Abstract: We prove a generalization of a so called "invariance principle" for partially hyperbolic diffeomorphisms: if an invariant probability measure has all its center Lyapunov exponents equal to zero then the measure admits a center disintegration that is invariant by stable and unstable holonomies. This was known for systems admitting a foliation by compact center leaves, and we extend it to a larger class which contains discretized Anosov flows. We use our result to classify measures of maximal entropy and study physical measures for perturbations of the time-one map of Anosov flows.

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References (45)
  1. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.
  2. Holonomy invariance: rough regularity and applications to Lyapunov exponents. Astérisque 358 (2013), 13–74.
  3. A. Avila and M. Viana. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181 (2010), 115–189.
  4. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. J. Eur. Math. Soc. 17 (2015), 1435,1462.
  5. Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent. Invent. Math. 231 (2023), 851–927.
  6. Collapsed anosov flows and self orbit equivalences. Preprint arXiv:2008.06547.
  7. C. Bonatti and L. J. Díaz. Persistent nonhyperbolic transitive diffeomorphisms. Annals of Math. 143 (1996), 357–396.
  8. C. Bonatti and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.
  9. R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes in Math. 470. Springer Verlag, 1975.
  10. A. Brown. Smoothness of stable holonomies inside center-stable manifolds. Ergod. Th & Dynam. Sys. (2021), 1–26.
  11. C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT actions on manifolds by lattices in Lie groups. Compos. Math. 158 (2022), 529–549.
  12. C. Butler. Rigidity of equality of Lyapunov exponents for geodesic flows. J. Differential Geom. 109 (2018), 39–79.
  13. A dichotomy for measures of maximal entropy near time-one maps of transitive anosov flows. Ann. Sci. Éc. Norm. Supér. 55 (2022), 969–1002.
  14. Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), 1687–1706.
  15. Equilibrium states in dynamical systems via geometric measure theory. Bulletin Amer. Math. Soc. 56 (2019), 569–610.
  16. Empirical measures of partially hyperbolic attractors. Communications in Mathematical Physics 375 (2020), 725–764.
  17. P. Didier. Stability of accessibility. Ergod. Th. & Dyanm. Sys. 23 (2003), 1717–1731.
  18. D. Dolgopyat. On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155 (2004), 389–449.
  19. T. Fisher and B. Hasselblatt. Hyperbolic flows. Zür. Lect. Adv. Math., EMS press, 2019.
  20. On the hybrid control of metric entropy for dominated splittings. Discrete & Contin. Dyn. Syst. 38 (2018), 5011.
  21. Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle. Invent. Math. 172 (2008), 353–381.
  22. Unstable entropies and variational principle for partially hyperbolic diffeomorphisms. Advances in Mathematics 321 (2017), 31–68.
  23. A new criterion of physical measures for partially hyperbolic diffeomorphisms. Trans. Amer. Math. Soc. 373 (2020), 385–417.
  24. F. Ledrappier. Positivity of the exponent for stationary sequences of matrices. Lect. Notes in Math. 1186 (1986), 56–73.
  25. F. Ledrappier and J.-M. Strelcyn. A proof of the estimation from below in pesin’s entropy formula. Ergod. Th & Dynam. Sys. 2 (1982), 203–219.
  26. F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. 122 (1985), 540–574.
  27. F. Ledrappier and J.-S. Xie. Vanishing transverse entropy in smooth ergodic theory. Ergod. Th & Dynam. Sys. 31 (2011), 1229–1235.
  28. G. A. Margulis. Certain measures associated with u-flows on compact manifolds. Functional Analysis and Its Applications 4 (1970), 55–67.
  29. S. Martinchich. Global stability of discretized anosov flows. Preprint arXiv:2204.03825.
  30. D. Obata and M. Poletti. Positive exponents for random products of conservative surface diffeomorphisms and some skew products. J. Dynam. Differential Equations 34 (2022), 2405–2428.
  31. J. C. Oxtoby. Measure and category, volume 2 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1980.
  32. Gibbs measures for partially hyperbolic attractors. Ergod. Th & Dynam. Sys. 2 (1982), 417–438.
  33. M. Poletti. Stably positive lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Contin. Dyn. Syst. 38 (2018.
  34. Hölder foliations. Duke Math. J. 86 (1997), 517–546.
  35. Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Th & Dynam. Sys. 32 (2012), 825–839.
  36. V. Rokhlin. On the fundamental ideas of measure theory. Mat. Sbornik N.S (1949), 107–150. and Amer. Math. Soc. Translation 1952 (1952), 55 pp.
  37. D. Ruelle. A measure associated with Axiom A attractors. Amer. J. Math. 98 (1976), 619–654.
  38. Y. Sinai. Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27 (1972), 21–69.
  39. A. Tahzibi and J. Yang. Invariance principle and rigidity of high entropy measures. Trans. Amer. Math. Soc. 371 (2019), 1231–1251.
  40. A. Tahzibi and J. Zhang. Disintegrations of non-hyperbolic ergodic measures along the center foliation of DA maps. Preprint arXiv:2206.08534.
  41. M. Viana. Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. Annals of Math. 167 (2008), 643–680.
  42. M. Viana and J. Yang. Physical measures and absolute continuity for one-dimensional center direction. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), 845–877.
  43. The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. 15 (2013), 2043–2060.
  44. A. Wilkinson. The cohomological equation for partially hyperbolic diffeomorphisms. Astérisque 358 (2013), 75–165.
  45. J. Yang. Entropy along expanding foliations. Advances in Mathematics 389 (2021), 107893.
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