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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

Hans Koch 1,

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.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.  Google Scholar

[7]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, AMS Colloquium Publications, 31, 1974.  Google Scholar

[8]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[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.  Google Scholar

[10]

H. Koch, Existence of Critical Invariant Tori, Erg. Theor. Dyn. Syst., 28 (2008), 1879-1894. doi: 10.1017/S0143385708000199.  Google Scholar

[11]

R. S. MacKay, Renormalisation in Area Preserving Maps, Thesis, Princeton, 1982. World Scientific, London, 1993. doi: 10.1142/9789814354462.  Google Scholar

[12]

R. S. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187. doi: 10.1088/0951-7715/5/1/007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Stirnemann, Renormalization for Golden Circles, Comm. Math. Phys., 152 (1993), 369-431. doi: 10.1007/BF02098303.  Google Scholar

[17]

A. Stirnemann, Towards an existence proof of mackay's fixed point, Comm. Math. Phys., 188 (1997), 723-735. doi: 10.1007/s002200050185.  Google Scholar

[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.  Google Scholar

[19]

Ada Reference Manual, ISO/IEC 8652:2012(E), available e.g. at http://www.ada-auth.org/arm.html., ().   Google Scholar

[20]

The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating-Point Arithmetic,, ANSI/IEEE Std 754-2008., (): 754.   Google Scholar

[21]

A free-software compiler for the Ada programming language, which is part of the GNU Compiler Collection,, see http://gcc.gnu.org/., ().   Google Scholar

[22]

The MPFR library for multiple-precision floating-point computations with correct rounding, see, http://www.mpfr.org/., ().   Google Scholar

[23]

The computer programs are available, at, ftp://ftp.ma.utexas.edu/pub/papers/koch/maps-spec/index.html., ().   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.  Google Scholar

[7]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, AMS Colloquium Publications, 31, 1974.  Google Scholar

[8]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[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.  Google Scholar

[10]

H. Koch, Existence of Critical Invariant Tori, Erg. Theor. Dyn. Syst., 28 (2008), 1879-1894. doi: 10.1017/S0143385708000199.  Google Scholar

[11]

R. S. MacKay, Renormalisation in Area Preserving Maps, Thesis, Princeton, 1982. World Scientific, London, 1993. doi: 10.1142/9789814354462.  Google Scholar

[12]

R. S. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187. doi: 10.1088/0951-7715/5/1/007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Stirnemann, Renormalization for Golden Circles, Comm. Math. Phys., 152 (1993), 369-431. doi: 10.1007/BF02098303.  Google Scholar

[17]

A. Stirnemann, Towards an existence proof of mackay's fixed point, Comm. Math. Phys., 188 (1997), 723-735. doi: 10.1007/s002200050185.  Google Scholar

[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.  Google Scholar

[19]

Ada Reference Manual, ISO/IEC 8652:2012(E), available e.g. at http://www.ada-auth.org/arm.html., ().   Google Scholar

[20]

The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating-Point Arithmetic,, ANSI/IEEE Std 754-2008., (): 754.   Google Scholar

[21]

A free-software compiler for the Ada programming language, which is part of the GNU Compiler Collection,, see http://gcc.gnu.org/., ().   Google Scholar

[22]

The MPFR library for multiple-precision floating-point computations with correct rounding, see, http://www.mpfr.org/., ().   Google Scholar

[23]

The computer programs are available, at, ftp://ftp.ma.utexas.edu/pub/papers/koch/maps-spec/index.html., ().   Google Scholar

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