On hyperbolicity in the renormalization of near-critical area-preserving maps

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December 2016, 36(12): 7029-7056. doi: 10.3934/dcds.2016106

On hyperbolicity in the renormalization of near-critical area-preserving maps

1.	Department of Mathematics, The University of Texas at Austin, Austin, TX 78712

Received February 2016 Revised June 2016 Published October 2016

Abstract
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We consider MacKay's renormalization operator for pairs of area-preserving maps, near the fixed point obtained in [1]. Of particular interest is the restriction $\mathfrak{R}_{0}$ of this operator to pairs that commute and have a zero Calabi invariant. We prove that a suitable extension of $\mathfrak{R}_{0}^{3}$ is hyperbolic at the fixed point, with a single expanding direction. The pairs in this direction are presumably commuting, but we currently have no proof for this. Our analysis yields rigorous bounds on various ``universal'' quantities, including the expanding eigenvalue.

Keywords: invariant circle, hyperbolicity., renormalization, Area-preserving maps.

Mathematics Subject Classification: Primary: 37E20; Secondary: 37F2.

Citation: Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106

References:

[1]	G. Arioli and H. Koch, The critical renormalization fixed point for commuting pairs of area-preserving maps, Comm. Math. Phys., 295 (2010), 415-429. doi: 10.1007/s00220-009-0922-1.
[2]	R. de la Llave and A. Olvera, The obstruction criterion for non-existence of invariant circles and renormalization, Nonlinearity, 19 (2006), 1907-1937. doi: 10.1088/0951-7715/19/8/008.
[3]	J.-P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc., 47 (1984), 1-121. doi: 10.1090/memo/0289.
[4]	C. Falcolini and R. de la Llave, A rigorous partial justification of Greene's criterion, J. Stat. Phys., 67 (1992), 609-643. doi: 10.1007/BF01049722.
[5]	D. Gaidashev, T. Johnson and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, Preprint, arXiv:1205.0826 (2012). doi: 10.1215/00127094-3165327.
[6]	J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.
[7]	E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, AMS Colloquium Publications, 31, 1974.
[8]	H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.
[9]	H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dynam. Systems, 11 (2004), 881-909. doi: 10.3934/dcds.2004.11.881.
[10]	H. Koch, Existence of Critical Invariant Tori, Erg. Theor. Dyn. Syst., 28 (2008), 1879-1894. doi: 10.1017/S0143385708000199.
[11]	R. S. MacKay, Renormalisation in Area Preserving Maps, Thesis, Princeton, 1982. World Scientific, London, 1993. doi: 10.1142/9789814354462.
[12]	R. S. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187. doi: 10.1088/0951-7715/5/1/007.
[13]	A. Olvera and C. Simó, An obstruction method for the destruction of invariant curves, Physica D, 26 (1987), 181-192. doi: 10.1016/0167-2789(87)90222-3.
[14]	S. Ostlund, D. Rand, J. Sethna and E. Siggia, Universal transition from quasiperiodicity to chaos in dissipative systems, Phys. Rev. Lett., 49 (1982), 132-135. doi: 10.1103/PhysRevLett.49.132.
[15]	S. J. Shenker and L. P. Kadanoff, Critical behaviour of KAM surfaces. I Empirical results, J. Stat. Phys., 27 (1982), 631-656. doi: 10.1007/BF01013439.
[16]	A. Stirnemann, Renormalization for Golden Circles, Comm. Math. Phys., 152 (1993), 369-431. doi: 10.1007/BF02098303.
[17]	A. Stirnemann, Towards an existence proof of mackay's fixed point, Comm. Math. Phys., 188 (1997), 723-735. doi: 10.1007/s002200050185.
[18]	M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes Etudes Sci., 96 (2002), 1-41. doi: 10.1007/s10240-003-0007-1.
[19]	Ada Reference Manual, ISO/IEC 8652:2012(E) available e.g. at http://www.ada-auth.org/arm.html.
[20]	The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-2008.
[21]	A free-software compiler for the Ada programming language, which is part of the GNU Compiler Collection, see http://gcc.gnu.org/.
[22]	The MPFR library for multiple-precision floating-point computations with correct rounding, see http://www.mpfr.org/.
[23]	The computer programs are available, at ftp://ftp.ma.utexas.edu/pub/papers/koch/maps-spec/index.html.

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References:

[1]	G. Arioli and H. Koch, The critical renormalization fixed point for commuting pairs of area-preserving maps, Comm. Math. Phys., 295 (2010), 415-429. doi: 10.1007/s00220-009-0922-1.
[2]	R. de la Llave and A. Olvera, The obstruction criterion for non-existence of invariant circles and renormalization, Nonlinearity, 19 (2006), 1907-1937. doi: 10.1088/0951-7715/19/8/008.
[3]	J.-P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc., 47 (1984), 1-121. doi: 10.1090/memo/0289.
[4]	C. Falcolini and R. de la Llave, A rigorous partial justification of Greene's criterion, J. Stat. Phys., 67 (1992), 609-643. doi: 10.1007/BF01049722.
[5]	D. Gaidashev, T. Johnson and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, Preprint, arXiv:1205.0826 (2012). doi: 10.1215/00127094-3165327.
[6]	J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.
[7]	E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, AMS Colloquium Publications, 31, 1974.
[8]	H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.
[9]	H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dynam. Systems, 11 (2004), 881-909. doi: 10.3934/dcds.2004.11.881.
[10]	H. Koch, Existence of Critical Invariant Tori, Erg. Theor. Dyn. Syst., 28 (2008), 1879-1894. doi: 10.1017/S0143385708000199.
[11]	R. S. MacKay, Renormalisation in Area Preserving Maps, Thesis, Princeton, 1982. World Scientific, London, 1993. doi: 10.1142/9789814354462.
[12]	R. S. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187. doi: 10.1088/0951-7715/5/1/007.
[13]	A. Olvera and C. Simó, An obstruction method for the destruction of invariant curves, Physica D, 26 (1987), 181-192. doi: 10.1016/0167-2789(87)90222-3.
[14]	S. Ostlund, D. Rand, J. Sethna and E. Siggia, Universal transition from quasiperiodicity to chaos in dissipative systems, Phys. Rev. Lett., 49 (1982), 132-135. doi: 10.1103/PhysRevLett.49.132.
[15]	S. J. Shenker and L. P. Kadanoff, Critical behaviour of KAM surfaces. I Empirical results, J. Stat. Phys., 27 (1982), 631-656. doi: 10.1007/BF01013439.
[16]	A. Stirnemann, Renormalization for Golden Circles, Comm. Math. Phys., 152 (1993), 369-431. doi: 10.1007/BF02098303.
[17]	A. Stirnemann, Towards an existence proof of mackay's fixed point, Comm. Math. Phys., 188 (1997), 723-735. doi: 10.1007/s002200050185.
[18]	M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes Etudes Sci., 96 (2002), 1-41. doi: 10.1007/s10240-003-0007-1.
[19]	Ada Reference Manual, ISO/IEC 8652:2012(E) available e.g. at http://www.ada-auth.org/arm.html.
[20]	The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-2008.
[21]	A free-software compiler for the Ada programming language, which is part of the GNU Compiler Collection, see http://gcc.gnu.org/.
[22]	The MPFR library for multiple-precision floating-point computations with correct rounding, see http://www.mpfr.org/.
[23]	The computer programs are available, at ftp://ftp.ma.utexas.edu/pub/papers/koch/maps-spec/index.html.

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