Atomic disintegrations for partially hyperbolic diffeomorphisms

Homburg, Ale

doi:10.1090/proc/13509

Atomic disintegrations for partially hyperbolic diffeomorphisms
HTML articles powered by AMS MathViewer

by Ale Jan Homburg PDF

Proc. Amer. Math. Soc. 145 (2017), 2981-2996 Request permission

Abstract:

Shub and Wilkinson and Ruelle and Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of circles. The foliation is not absolutely continuous; in fact, Ruelle and Wilkinson established that the disintegration of volume along central leaves is atomic. We show that in such a class of volume preserving diffeomorphisms the disintegration of volume along central leaves is a single delta measure. We also formulate a general result for conservative three dimensional skew product like diffeomorphisms on circle bundles, providing conditions for delta measures as disintegrations of the smooth invariant measure.

References

Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Alexandre T. Baraviera and Christian Bonatti, Removing zero Lyapunov exponents, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1655–1670. MR 2032482, DOI 10.1017/S0143385702001773
Luis Barreira and Yakov Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR 2348606, DOI 10.1017/CBO9781107326026
G. R. Belickiĭ, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Funkcional. Anal. i Priložen. 7 (1973), no. 4, 17–28 (Russian). MR 0331437
Doris Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, J. Mod. Dyn. 7 (2013), no. 4, 565–604. MR 3177773, DOI 10.3934/jmd.2013.7.565
Christian Bonatti and Sylvain Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), no. 1, 33–104 (French, with English and French summaries). MR 2090361, DOI 10.1007/s00222-004-0368-1
Christian Bonatti, Lorenzo J. Díaz, and Raúl Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu 1 (2002), no. 4, 513–541. MR 1954435, DOI 10.1017/S1474748002000142
Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774
Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, DOI 10.1007/BF02810585
Christian Bonatti and Amie Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology 44 (2005), no. 3, 475–508. MR 2122214, DOI 10.1016/j.top.2004.10.009
Keith Burns, Dmitry Dolgopyat, Yakov Pesin, and Mark Pollicott, Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn. 2 (2008), no. 1, 63–81. MR 2366230, DOI 10.3934/jmd.2008.2.63
Keith Burns and Amie Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), no. 1, 451–489. MR 2630044, DOI 10.4007/annals.2010.171.451
Hans Crauel, Extremal exponents of random dynamical systems do not vanish, J. Dynam. Differential Equations 2 (1990), no. 3, 245–291. MR 1066618, DOI 10.1007/BF01048947
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
John Milnor, Fubini foiled: Katok’s paradoxical example in measure theory, Math. Intelligencer 19 (1997), no. 2, 30–32. MR 1457445, DOI 10.1007/BF03024428
Yakov B. Pesin, Lectures on partial hyperbolicity and stable ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. MR 2068774, DOI 10.4171/003
Charles Pugh and Michael Shub, Stable ergodicity, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 1, 1–41. With an appendix by Alexander Starkov. MR 2015448, DOI 10.1090/S0273-0979-03-00998-4
Charles Pugh, Michael Shub, and Amie Wilkinson, Hölder foliations, Duke Math. J. 86 (1997), no. 3, 517–546. MR 1432307, DOI 10.1215/S0012-7094-97-08616-6
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures, Creation of blenders in the conservative setting, Nonlinearity 23 (2010), no. 2, 211–223. MR 2578476, DOI 10.1088/0951-7715/23/2/001
F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), no. 2, 353–381. MR 2390288, DOI 10.1007/s00222-007-0100-z
Federico Rodriguez Hertz, Jana Rodriguez Hertz, and Raúl Ures, Partially hyperbolic dynamics, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011. 28$^\textrm {o}$ Colóquio Brasileiro de Matemática. [28th Brazilian Mathematics Colloquium]. MR 2828094
David Ruelle and Amie Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys. 219 (2001), no. 3, 481–487. MR 1838747, DOI 10.1007/s002200100420
Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057, DOI 10.1007/s002229900035
Marcelo Viana and Jiagang Yang, Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 5, 845–877. MR 3103173, DOI 10.1016/j.anihpc.2012.11.002