On the geometry of period doubling invariant sets for area-preserving maps

Denis Gaidashev; Dan Lilja

doi:10.3934/dcds.2025082

Article Contents

2025, Volume 45, Issue 12: 5036-5074. Doi: 10.3934/dcds.2025082

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On the geometry of period doubling invariant sets for area-preserving maps

Denis Gaidashev^1, , and
Dan Lilja^2,

1.
Department of Mathematics, Uppsala University, Uppsala, Sweden
2.
Consid, Sweden

^*Corresponding author: Denis Gaidashev
^*Corresponding author: Denis Gaidashev

Received: April 2023

Revised: May 2025

Early access: June 11, 2025

Published online: June 11, 2025

The first author is supported by Vetenskapsrådet project grant 2022-05046.

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Abstract

The geometry of the period doubling Cantor sets of strongly dissipative infinitely renormalizable Hénon-like maps has been shown to be unbounded by M. Lyubich, M. Martens, and A. de Carvalho, although the measure of unbounded "spots" in the Cantor set has been demonstrated to be zero.

We show that an even more extreme situation takes places for infinitely renormalizable area-preserving Hénon-like maps: Both bounded and unbounded geometries exist on subsets of positive measure.

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Mathematics Subject Classification: Primary: 37E20, 37E30, 37E40; Secondary: 37J11.

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Figure 2.1. The ratchet phenomenon for a positive twist map. A negative twist map reverses the signs

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Figure 3.1. The renormalization microscope

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Figure 6.1. Unbounded geometry near the tip

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References

[1]	L. Arnold, Random Dynamical Systems, Springer, 2003. doi: 10.1007/978-3-662-12878-7.
[2]	A. Avila and M. Lyubich, The full renormalization horseshoe for unimodal maps of higher degree: Exponential contraction along hybrid classes, Publications Mathématiques de L'IHÉS, 114 (2011), 171-223. doi: 10.1007/s10240-011-0034-2.
[3]	G. Benettin, C. Cercignani, L. Galgani and A. Giorgilli, Universal properties in conservative dynamical systems, Lettere al Nuovo Cimento (1971-1985), 28 (1980), 1-4. doi: 10.1007/BF02776065.
[4]	T. C. Bountis, Period doubling bifurcations and universality in conservative systems, Physica D: Nonlinear Phenomena, 3 (1981), 577-589. doi: 10.1016/0167-2789(81)90041-5.
[5]	A. de Carvalho, M. Lyubich and M. Martens, Renormalization in the Hénon family, I: Universality but non-rigidity, Journal of Statistical Physics, 121 (2005), 611-669. doi: 10.1007/s10955-005-8668-4.
[6]	P. Collet, J.-P. Eckmann and H. Koch, On universality for area-preserving maps of the plane, Phys. D, 3 (1981), 457-467. doi: 10.1016/0167-2789(81)90033-6.
[7]	P. Collet, J.-P. Eckmann and H. Koch, Period doubling bifurcations for families of maps on $\mathbb{R}^{n}$, J. Statist. Phys., 25 (1981), 1-14. doi: 10.1007/BF01008475.
[8]	P. Coullet and C. Tresser, Itérations d'endomorphismes et groupe de renormalisation, J. Phys. Colloque C, 39 (1978), C5-25-C5-28. doi: 10.1051/jphyscol:1978513.
[9]	E. de Faria, W. de Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math. (2), 164 (2006), 731-824. doi: 10.4007/annals.2006.164.731.
[10]	W. de Melo and S. van Strien, One-Dimensional Dynamics, vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.
[11]	B. Derrida and Y. Pomeau, Feigenbaum's ratios of two-dimensional area preserving maps, Physics Letters A, 80 (1980), 217-219. doi: 10.1016/0375-9601(80)90003-1.
[12]	J.-P. Eckmann, H. Koch and P. Wittwer, Existence of a fixed point of the doubling transformation for area-preserving maps of the plane, Phys. Rev. A, 26 (1982), 720-722. doi: 10.1103/PhysRevA.26.720.
[13]	J.-P. Eckmann, H. Koch and P. Wittwer, A Computer-Assisted Proof of Universality for Area-Preserving Maps, Mem. Amer. Math. Soc., 47 (1984), ⅵ+122. doi: 10.1090/memo/0289.
[14]	M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., 19 (1978), 25-52. doi: 10.1007/BF01020332.
[15]	M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys., 21 (1979), 669-706. doi: 10.1007/BF01107909.
[16]	D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: Hyperbolic sets, Nonlinearity, 22 (2009), 2487-2520. doi: 10.1088/0951-7715/22/10/010.
[17]	D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period-doubling: Stable sets, Journal of Modern Dynamics, 3 (2009), 555-587. doi: 10.3934/jmd.2009.3.555.
[18]	D. Gaidashev and T. Johnson, Spectral properties of renormalization for area-preserving maps, Discrete and Continuous Dynamical Systems, 36 (2016), 3651-3675. doi: 10.3934/dcds.2016.36.3651.
[19]	D. Gaidashev, T. Johnson and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, Duke Math. J., 165 (2016), 129-159. doi: 10.1215/00127094-3165327.
[20]	J. M. Gambaudo, S. van Strien and C. Tresser, Hénon-like maps with strange attractors: there exist $C^{\infty}$ Kupka-Smale diffeomorphisms on $S^{2}$ with neither sinks nor sources, Nonlinearity, 2 (1989), 287-304. doi: 10.1088/0951-7715/2/2/005.
[21]	C. Golé, Symplectic Twist Maps. Global Variational Techniques, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812810762.
[22]	P. E. Hazard, Hénon-like maps with arbitrary stationary combinatorics, Ergodic Theory and Dynamical Systems, 31 (2011), 1391-1443. doi: 10.1017/S0143385710000398.
[23]	P. E. Hazard, M. Lyubich and M. Martens, Renormalizable Hénon-like maps and unbounded geometry, Nonlinearity, 25 (2012), 397-420. doi: 10.1088/0951-7715/25/2/397.
[24]	R. Helleman, Self-generated chaotic behavior in nonlinear mechanics, in Universality in Chaos (ed. P. Cvitanović), 2nd edition, New York: Routledge, 1989, chapter 41,420-488. doi: 10.1201/9780203734636-43.
[25]	M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publications Mathématiques de l'IHÉS, 49 (1979), 5-233, http://www.numdam.org/item/PMIHES_1979__49__5_0. doi: 10.1007/BF02684798.
[26]	M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture, Annals of Mathematics. Second Series, 149 (1999), 319-420. doi: 10.2307/120968.
[27]	M. Lyubich and M. Martens, Probabilistic universality in two-dimensional dynamics, Comm. Math. Phys., 383 (2021), 1295–1359. arXiv e-prints. doi: 10.1007/s00220-021-03960-z.
[28]	M. Lyubich and M. Martens, Renormalization of Hénon maps, in Dynamics, Games and Science I: DYNA 2008, in Honor of Maurício Peixoto and David Rand, University of Minho, Braga, Portugal, September 8-12, 2008 (eds. M. M. Peixoto, A. A. Pinto and D. A. Rand), Springer Berlin Heidelberg, Berlin, Heidelberg, 2011,597-618. doi: 10.1007/978-3-642-11456-4_37.
[29]	M. Lyubich and M. Martens, Renormalization in the Hénon family, Ⅱ: The heteroclinic web, Inventiones Mathematicae, 186 (2011), 115-189. doi: 10.1007/s00222-011-0316-9.
[30]	M. Martens, The periodic points of renormalization, Annals of Mathematics, 147 (1998), 543-584. doi: 10.2307/120959.
[31]	C. T. McMullen, Renormalization and 3-Manifolds Which Fiber Over the Circle (AM-142), Princeton University Press, 1996. doi: 10.1515/9781400865178.
[32]	M. Misiurewicz, Structure of mappings of an interval with zero entropy, Publications Mathématiques de l'IHÉS, 53 (1981), 5-16, http://www.numdam.org/item/PMIHES_1981__53__5_0. doi: 10.1007/BF02698685.
[33]	V. V. Petrov, On the strong law of large numbers for stationary sequences, Dokl. Akad. Nauk. SSSR, 213 (1973), 42-44.
[34]	Y. G. Sinai, Dynamical Systems II. Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics, Springer-Verlag, Berlin, New-York, 1997.
[35]	D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, AMS Centennial Publicaitons.
[36]	C. Tresser and P. Coullet, Itérations d'endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A577-A580.
[37]	E. B. Vul, Y. G. Sinai and K. M. Khanin, Feigenbaum universality and the thermodynamic formalism, Russian Mathematical Surveys, 39 (1984), 1-40. doi: 10.1070/RM1984v039n03ABEH003162.
[38]	M. Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps, Communications in Mathematical Physics, 218 (2001), 537-568. doi: 10.1007/PL00005561.