The Lorenz Renormalization Conjecture

References

• V. S. Afraimovich, V. V. Bykov, L. P. Shilnikov. “On the Origin and Structure of the Lorenz Attractor.” Soviet Phys Doklady. 22 (1977), 253–255.
   Google Scholar
• A. Arneodo, P. Coullet, C. Tresser. “A Possible New Mechanism for the Onset of Turbulence.” Phys Lett A. 81: 4 (1981), 197–201.
   Web of Science ®Google Scholar
• V. I. Arnolˊd. “Small Denominators. I. Mapping the Circle Onto Itself.” Izv Akad Nauk SSSR Ser Mat. 25 (1961), 21–86.
   Google Scholar
• A. Avila, M. Lyubich. “The Full Renormalization Horseshoe For Unimodal Maps of Higher Degree: Exponential Contraction Along Hybrid Classes.” Publ Math Inst Hautes Etudes Sci. 114 (2011), 171–223.
   Google Scholar
• V. V. M. S. Chandramouli, M. Martens, W. de Melo, C. P. Tresser. “Chaotic Period Doubling.” Ergod Theory Dyn Syst. 29: 2 (2009), 381–418.
   Google Scholar
• P., J.-P. Collet, Eckmann, H. Koch. “Period Doubling Bifurcations for Families of Maps on Rn.” J Statist Phys. 25: 1 (1981), 1–14.
   Web of Science ®Google Scholar
• P. Coullet and C. Tresser. “Itérations d'endomorphismes et groupe de renormalisation.” C R Acad Sci Paris Sér A-B. 287: 7 (1978), A577–A580.
   Google Scholar
• A. De Carvalho, M. Lyubich, M. Martens. “Renormalization in the Hénon Family. I. Universality but Non-rigidity.” J Stat Phys. 121: 5–6 (2005), 611–669.
   Web of Science ®Google Scholar
• A. Douady, J. H. Hubbard. “A Proof of Thurston's Topological Characterization of Rational Functions.” Acta Math. 171 (1993), 263–297.
   Google Scholar
• J.-P. Eckmann, H. Koch, P. Wittwer. “A Computer-Assisted Proof of Universality for Area-Preserving Maps.” Mem Amer Math Soc. 47: 289 (1984), vi+122.
   Web of Science ®Google Scholar
• de Faria, E. “Proof of Universality for Critical Circle Mappings.” Thesis (Ph.D.), ProQuest LLC, Ann Arbor, MI, City University of New York, 1992.
   Google Scholar
• M. J. Feigenbaum. “Quantitative Universality for a Class of Nonlinear Transformations.” J Statist Phys. 19: 1 (1978), 25–52.
   Web of Science ®Google Scholar
• L. Fousse, G. Hanrot, V. Lef_evre, P. Pélissier, P. Zimmermann. “MPFR: A Multiple-precision Binary Floating-point Library with Correct Rounding.” ACM Trans Math Softw. 33: 2 (2007).
   Web of Science ®Google Scholar
• D. Gaidashev, T. Johnson, M. Martens. “Rigidity for Infinitely Renormalizable Area-Preserving Maps.” Duke Math J. 165: 1 (2016), 129–159.
   Web of Science ®Google Scholar
• J. Guckenheimer, R. F. Williams. “Structural Stability of Lorenz Attractors.” Inst Hautes Études Sci Publ Math. 50 (1979), 59–72.
   Google Scholar
• G. Guennebaud, B. Jacob. “Eigen v3.” Available at (http://eigen.tuxfamily.org), 2010.
   Google Scholar
• M. Herman. “Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations.” Inst Hautes Études Sci Publ Math. 49 (1979), 5–233.
   Google Scholar
• M. W. Hirsch, C. C. Pugh, M. Shub. Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Berlin, New York: Springer-Verlag, 1977.
   Google Scholar
• P. Holoborodko. “MPFR C++.” Available at (http://www.holoborodko.com/pavel/mpfr/), 2008.
   Google Scholar
• J. H. Hubbard, D. Schleicher. “The Spider Algorithm.” Proc. Sympos. Appl. Math., Vol. 49. Providence, RI: Amer. Math. Soc., 1994.
   Google Scholar
• K. Khanin, A. Teplinsky. “Renormalization Horseshoe and Rigidity for Circle Diffeomorphisms With Breaks.” Comm Math Phys. 320: 2 (2013), 347–377.
   Web of Science ®Google Scholar
• G. Levin, G. Światek. “Universality of Critical Circle Covers.” Comm Math Phys. 228: 2 (2002), 371–399.
   Web of Science ®Google Scholar
• P. S. Linsay. “Period Doubling and Chaotic Behavior in a Driven Anharmonic Oscillator.” Phys Rev Lett. 47: 19 (1981), 1349–1352.
   Web of Science ®Google Scholar
• E. N. Lorenz. “Deterministic Nonperiodic Flow.” J Atmospheric Sci. 20 (1963), 130–141.
   Web of Science ®Google Scholar
• M. Lyubich. “Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture.” Ann Math. (2) 149: 2 (1999), 319–420.
   Web of Science ®Google Scholar
• M. Martens, W. de Melo. “Universal Models for Lorenz Maps.” Ergod Theory Dyn Syst.” 21: 3 (2001), 833–860.
   Google Scholar
• M. Martens, L. Palmisano. “Rigidity foliations.” Available at (arXiv:1704.06328), 2017.
   Google Scholar
• M. Martens, L. Palmisano, B. Winckler. “The Rigidity Conjecture.” Indagationes Math. (2017).
   Web of Science ®Google Scholar
• M. Martens, B. Winckler. “On the Hyperbolicity of Lorenz Renormalization.” Comm Math Phys. 325: 1 (2014), 185–257.
   Web of Science ®Google Scholar
• M. Martens, B. Winckler. “Instability of Renormalization.” Available at (arXiv:1609.04473), 2017.
   Google Scholar
• M. Martens, B. Winckler. “Physical Measures for Infinitely Renormalizable Lorenz Maps.” Ergod Theory Dyn Syst. 38: 2 (2018), 717–738.
   Google Scholar
• J. Maurer, A. Libchaber. “Rayleigh–Bénard Experiment in Liquid Helium; Frequency Locking and the Onset Of Turbulence.” J Phys Lett. 40: 16 (1979), 419–423.
   Google Scholar
• C. T. McMullen. Renormalization and 3-manifolds which fiber over the circle, Ann. of Math. Studies, Vol. 142. Princeton, NJ: Princeton University Press, 1996.
   Google Scholar
• W. de Melo, S. van Strien. One-dimensional Dynamics, Vol. 25, Berlin: Springer-Verlag, 1993.
   Google Scholar
• J. Milnor. “Fubini Foiled: Katok's Paradoxical Example in Measure Theory.” Math Intelligencer. 19: 2 (1997), 30–32.
   Web of Science ®Google Scholar
• D. Sullivan. “Bounds, quadratic di_erentials, and renormalization conjectures.” American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), pp. 417–466. Providence, RI: Amer. Math. Soc.,1992.
   Google Scholar
• B. Winckler. “The Lorenz Renormalization Conjecture – Supplementary Material.” Available at (https://github.com/b4winckler/src-lorenz-renorm-conj), 2018.
   Google Scholar
• M. Yampolsky. “Renormalization Horseshoe for Critical Circle Maps.” Comm Math Phys. 240: 1–2 (2003), 75–96.
   Web of Science ®Google Scholar