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

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.
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.

show all references

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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