# American Institute of Mathematical Sciences

July  2016, 36(7): 3651-3675. doi: 10.3934/dcds.2016.36.3651

## Spectral properties of renormalization for area-preserving maps

 1 Department of Mathematics, Uppsala University, Uppsala, Sweden 2 Fraunhofer-Chalmers Research Centre for Industrial Mathematics, SE-412 88 Gothenburg, Sweden

Received  December 2014 Revised  November 2015 Published  March 2016

Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point.
Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate.
This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point.
In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.
Citation: Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
##### References:
 [1] J. J. Abad and H. Koch, Renormalization and periodic orbits for Hamiltonian flows, Comm. Math. Phys., 212 (2000), 371-394. doi: 10.1007/s002200000218. [2] J. J. Abad, H. Koch and P. Wittwer, A renormalization group for Hamiltonians: Numerical results}, Nonlinearity, 11 (1998), 1185-1194. doi: 10.1088/0951-7715/11/5/001. [3] G. Benettin et al, Universal properties in conservative dynamical systems, Lettere al Nuovo Cimento, 28 (1980), 1-4. [4] T. Bountis, Period doubling bifurcations and universality in conservative Systems, Physica, 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, J. Stat. Phys, 121 (2005), 611-669. doi: 10.1007/s10955-005-8668-4. [6] P. Collet, J.-P. Eckmann and H. Koch, Period doubling bifurcations for families of maps on $\mathbbR^n$, J. Stat. Phys., 3D (1980). [7] P. Collet, J.-P. Eckmann and H. Koch, On universality for area-preserving maps of the plane, Physica D, 3 (1981), 457-467. doi: 10.1016/0167-2789(81)90033-6. [8] B. Derrida and Y. Pomeau, Feigenbaum's ratios of two dimensional area preserving maps, Phys. Lett. A, 80 (1980), 217-219. doi: 10.1016/0375-9601(80)90003-1. [9] 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. [10] J.-P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Memoirs of the American Mathematical Society, 47 (1984), vi+122 pp. doi: 10.1090/memo/0289. [11] H. Epstein, New proofs of the existence of the Feigenbaum functions, Commun. Math. Phys., 106 (1986), 395-426. doi: 10.1007/BF01207254. [12] D. Gaidashev, Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori, Discrete Contin. Dyn. Syst., 13 (2005), 63-102. doi: 10.3934/dcds.2005.13.63. [13] D. Gaidashev, Period doubling renormalization for area-preserving maps and mild computer assistance in contraction mapping principle, Int. Journal of Bifurcations and Chaos, 21 (2011), 3217-3230. doi: 10.1142/S0218127411030477. [14] 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. [15] D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: Stable sets, J. Mod. Dyn., 3 (2009), 555-587. doi: 10.3934/jmd.2009.3.555. [16] D. Gaidashev, T. Johnson and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, Duke Mathematical Journal, 165 (2016), 129-159. doi: 10.1215/00127094-3165327. [17] D. Gaidashev and H. Koch, Renormalization and shearless invariant tori: Numerical results, Nonlinearity, 17 (2004), 1713-1722. doi: 10.1088/0951-7715/17/5/008. [18] D. Gaidashev and H. Koch, Period doubling in area-preserving maps: An associated one-dimensional problem, Ergod. Th. & Dyn. Sys., 31 (2011), 1193-1228. doi: 10.1017/S0143385710000283. [19] P. Hazard, Hénon-like maps with arbitrary stationary combinatorics, Ergod. Th. & Dynam. Sys., 31 (2011), 1391-1443. doi: 10.1017/S0143385710000398. [20] P. E. Hazard, M. Lyubich and M. Martens, Renormalisable Henon-like maps and unbounded geometry, Nonlinearity, 25 (2012), 397-420. doi: 10.1088/0951-7715/25/2/397. [21] R. H. G. Helleman, Self-generated chaotic behavior in nonlinear mechanics, in Fundamental Problems in Statistical Mechanics (ed. E. G. D. Cohen), North-Holland, Amsterdam, 1980, 165-233. [22] T. Johnson, No elliptic islands for the universal area-preserving map, Nonlinearity, 24 (2011), 2063-2078. doi: 10.1088/0951-7715/24/7/008. [23] K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231. doi: 10.1007/s00220-006-0125-y. [24] H. Koch, On the renormalization of Hamiltonian flows, and critical invariant tori, Discrete Contin. Dyn. Syst., 8 (2002), 633-646. doi: 10.3934/dcds.2002.8.633. [25] H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dyn. Syst., 11 (2004), 881-909. doi: 10.3934/dcds.2004.11.881. [26] H. Koch, Existence of critical invariant tori, Ergod. Th. & Dynam. Sys., 28 (2008), 1879-1894. doi: 10.1017/S0143385708000199. [27] S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544. doi: 10.1088/0951-7715/18/6/006. [28] M. Lyubich and M. Martens, Renormalization in the Hénon family, II: Homoclinic tangle, Invent. Math., 186 (2011), 115-189. doi: 10.1007/s00222-011-0316-9. [29] M. Lyubich and M. Martens, Probabilistic universality in two-dimensional dynamics, e-print arXiv:1106.5067, (2011). [30] Y. W. Nam, Renormalization for three-dimensional Hénon-like maps, e-print arXiv:1408.4289, (2014). [31] , Programs, available at , ().

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##### References:
 [1] J. J. Abad and H. Koch, Renormalization and periodic orbits for Hamiltonian flows, Comm. Math. Phys., 212 (2000), 371-394. doi: 10.1007/s002200000218. [2] J. J. Abad, H. Koch and P. Wittwer, A renormalization group for Hamiltonians: Numerical results}, Nonlinearity, 11 (1998), 1185-1194. doi: 10.1088/0951-7715/11/5/001. [3] G. Benettin et al, Universal properties in conservative dynamical systems, Lettere al Nuovo Cimento, 28 (1980), 1-4. [4] T. Bountis, Period doubling bifurcations and universality in conservative Systems, Physica, 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, J. Stat. Phys, 121 (2005), 611-669. doi: 10.1007/s10955-005-8668-4. [6] P. Collet, J.-P. Eckmann and H. Koch, Period doubling bifurcations for families of maps on $\mathbbR^n$, J. Stat. Phys., 3D (1980). [7] P. Collet, J.-P. Eckmann and H. Koch, On universality for area-preserving maps of the plane, Physica D, 3 (1981), 457-467. doi: 10.1016/0167-2789(81)90033-6. [8] B. Derrida and Y. Pomeau, Feigenbaum's ratios of two dimensional area preserving maps, Phys. Lett. A, 80 (1980), 217-219. doi: 10.1016/0375-9601(80)90003-1. [9] 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. [10] J.-P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Memoirs of the American Mathematical Society, 47 (1984), vi+122 pp. doi: 10.1090/memo/0289. [11] H. Epstein, New proofs of the existence of the Feigenbaum functions, Commun. Math. Phys., 106 (1986), 395-426. doi: 10.1007/BF01207254. [12] D. Gaidashev, Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori, Discrete Contin. Dyn. Syst., 13 (2005), 63-102. doi: 10.3934/dcds.2005.13.63. [13] D. Gaidashev, Period doubling renormalization for area-preserving maps and mild computer assistance in contraction mapping principle, Int. Journal of Bifurcations and Chaos, 21 (2011), 3217-3230. doi: 10.1142/S0218127411030477. [14] 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. [15] D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: Stable sets, J. Mod. Dyn., 3 (2009), 555-587. doi: 10.3934/jmd.2009.3.555. [16] D. Gaidashev, T. Johnson and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, Duke Mathematical Journal, 165 (2016), 129-159. doi: 10.1215/00127094-3165327. [17] D. Gaidashev and H. Koch, Renormalization and shearless invariant tori: Numerical results, Nonlinearity, 17 (2004), 1713-1722. doi: 10.1088/0951-7715/17/5/008. [18] D. Gaidashev and H. Koch, Period doubling in area-preserving maps: An associated one-dimensional problem, Ergod. Th. & Dyn. Sys., 31 (2011), 1193-1228. doi: 10.1017/S0143385710000283. [19] P. Hazard, Hénon-like maps with arbitrary stationary combinatorics, Ergod. Th. & Dynam. Sys., 31 (2011), 1391-1443. doi: 10.1017/S0143385710000398. [20] P. E. Hazard, M. Lyubich and M. Martens, Renormalisable Henon-like maps and unbounded geometry, Nonlinearity, 25 (2012), 397-420. doi: 10.1088/0951-7715/25/2/397. [21] R. H. G. Helleman, Self-generated chaotic behavior in nonlinear mechanics, in Fundamental Problems in Statistical Mechanics (ed. E. G. D. Cohen), North-Holland, Amsterdam, 1980, 165-233. [22] T. Johnson, No elliptic islands for the universal area-preserving map, Nonlinearity, 24 (2011), 2063-2078. doi: 10.1088/0951-7715/24/7/008. [23] K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231. doi: 10.1007/s00220-006-0125-y. [24] H. Koch, On the renormalization of Hamiltonian flows, and critical invariant tori, Discrete Contin. Dyn. Syst., 8 (2002), 633-646. doi: 10.3934/dcds.2002.8.633. [25] H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dyn. Syst., 11 (2004), 881-909. doi: 10.3934/dcds.2004.11.881. [26] H. Koch, Existence of critical invariant tori, Ergod. Th. & Dynam. Sys., 28 (2008), 1879-1894. doi: 10.1017/S0143385708000199. [27] S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544. doi: 10.1088/0951-7715/18/6/006. [28] M. Lyubich and M. Martens, Renormalization in the Hénon family, II: Homoclinic tangle, Invent. Math., 186 (2011), 115-189. doi: 10.1007/s00222-011-0316-9. [29] M. Lyubich and M. Martens, Probabilistic universality in two-dimensional dynamics, e-print arXiv:1106.5067, (2011). [30] Y. W. Nam, Renormalization for three-dimensional Hénon-like maps, e-print arXiv:1408.4289, (2014). [31] , Programs, available at , ().
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