A Numerical Study of Gibbs u-Measures for Partially Hyperbolic Diffeomorphisms on T3

References

• [Bonatti et al. 02] C. Bonatti, L. Díaz, and R. Ures. “Minimality of Strong Stable and Unstable Foliations for Partially Hyperbolic Diffeomorphisms.” J. Inst. Math. Jussieu 1:4 (2002), 513–541.
   Web of Science ®Google Scholar
• [Bonatti et al. 05] C. Bonatti, L. Díaz, and M. Viana. “Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective.” In Encyclopaedia of Mathematical Sciences, Jürg Fröhlich, S.P. Novikov, D. Ruelle (eds.), vol. 102. Mathematical Physics, III, xviii+384 pp. Berlin: Springer-Verlag, 2005.
   Google Scholar
• [Bonatti and Viana 00] C. Bonatti and M. Viana. “SRB Measures for Partially Hyperbolic Systems Whose Central Direction is Mostly Contracting.” Israel J. Math. 115 (2000), 157–193.
   Web of Science ®Google Scholar
• [Box and Muller 58] G. E. P. Box and M. E. Muller. “A Note on the Generation of Random Normal Deviates.” Annals Math. Stat. 29 (1958), 610–611.
   Google Scholar
• [Brin and Pesin 74] M. Brin and Y. Pesin. “Partially Hyperbolic Dynamical Systems.” Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.
   Google Scholar
• [Crovisier and Potrie 15] S. Crovisier and R. Potrie. Introduction to Partially Hyperbolic Dynamics. Lecture notes for a mini-course in School and Conference in Dynamical Systems, ICTP, Trieste, 2015.
   Google Scholar
• [Dolgopyat 01] D. Dolgopyat. Lectures on u-Gibbs States. Paper presented at Conference on Partially Hyperbolic Systems, Northwestern University, 2001.
   Google Scholar
• [Dolgopyat 04] D. Dolgopyat. “Limit Theorems for Partially Hyperbolic Systems.” Trans. Am. Math. Soc. 356:4 (2004), 1637–1689.
   Web of Science ®Google Scholar
• [Dolgopyat 04] D. Dolgopyat. “On Differentiability of SRB States for Partially Hyperbolic Systems.” Invent. Math. 155:2 (2004), 389–449.
   Web of Science ®Google Scholar
• [Ren et al. 17] Y. Ren, S. Gan, and P. Zhang. “Accessibility and Homology Bounded Strong Unstable Foliation for Anosov Diffeomorphisms on 3-Torus.” Acta Math. Sin. (Engl. Ser.) 33:1 (2017), 71–76.
   Web of Science ®Google Scholar
• [Gogolev 08] A. Gogolev, “Smooth Conjugacy of Anosov Diffeomorphisms on Higher-Dimensional Tori.” J. Mod. Dyn. 2:4 (2008), 645–700.
   Web of Science ®Google Scholar
• [Gogolev and Guysinsky 08] A. Gogolev and M. Guysinsky. “C1-Differentiable Conjugacy of Anosov Diffeomorphisms on Three Dimensional Torus.” Discrete Contin. Dyn. Syst. 22:1–2 (2008), 183–200.
   Web of Science ®Google Scholar
• [Hirsch et al. 77] M. Hirsch, C. Pugh, and M. Shub. “Invariant Manifolds.” In Lecture Notes, 583. Springer, 1977.
   Google Scholar
• [Jiang 12] M. Jiang. “Differentiating Potential Functions of SRB Measures on Hyperbolic Attractors.” Ergodic Theory Dynam. Systems 32:4 (2012), 1350–1369.
   Google Scholar
• [Mañé 78] R. Mañé. “Contributions to the Stability Conjecture.” Topology 17 (1978), 383–396.
   Web of Science ®Google Scholar
• [Park and Miller 88] S. K. Park and K. W. Miller. “Random Number Generators: Good Ones are Hard to Find.” Commun. ACM 31 (1988), 1192–1201.
   Web of Science ®Google Scholar
• [Pesin 04] Y. Pesin. Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lectures in Advanced Mathematics, vi+122 pp. Zürich: European Mathematical Society (EMS), 2004.
   Google Scholar
• [Pesin and Sinai 82] Y. Pesin and Y. Sinai. “Gibbs Measures for Partially Hyperbolic Attractors.” Ergodic Theory Dynam. Systems 2:3–4 (1982), 417–438.
   Google Scholar
• [Potrie 14] R. Potrie. “A Few Remarks on Partially Hyperbolic Diffeomorphisms of T3 Isotopic to Anosov.” J. Dynam. Differential Equations 26:3 (2014), 805–815.
   Web of Science ®Google Scholar
• [Pujals and Sambarino 06] E. Pujals and M. Sambarino. “A Sufficient Condition for Robustly Minimal Foliations.” Ergodic Theory Dynam. Systems 26:1 (2006), 281–289.
   Google Scholar
• [Ruelle 97] D. Ruelle. “Differentiation of SRB States.” Commun. Math. Phys. 187:1 (1997), 227–241.
   Web of Science ®Google Scholar
• [Ruelle 98] D. Ruelle. “Nonequilibrium Statistical Mechanics Near Equilibrium: Computing Higher-Order Terms.” Nonlinearity 11:1 (1998), 5–18.
   Web of Science ®Google Scholar
• [Smirnoff 39] N. Smirnoff. “On the Estimation of the Discrepancy Between Empirical Curves of Distribution for Two Independent Samples.” Bull. Math. Univ. Moscou 2:2 (1939), 16.
   Google Scholar
• [Yang XX] J. Yang. “Entropy Along Expanding Foliations.” arXiv:1601.05504v1.
   Google Scholar
• [Young 86] L.-S. Young. “Stochastic Stability of Hyperbolic Attractors.” Ergodic Theory Dynam. Systems 6:2 (1986), 311–319.
   Google Scholar
• [Young 02] L.-S. Young. “What Are SRB Measures, and Which Dynamical Systems have Them? Dedicated to David Ruelle and Yasha Sinai on the Occasion of Their 65th Birthdays.” J. Statist. Phys. 108:5–6 (2002), 733–754.
   Web of Science ®Google Scholar