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On hyperbolicity in the renormalization of near-critical area-preserving maps
1. | Department of Mathematics, The University of Texas at Austin, Austin, TX 78712 |
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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