American Institute of Mathematical Sciences

Advanced
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

[1]

Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
[2]

Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007
[3]

Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231
[4]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[5]

Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555
[6]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[7]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[8]

Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633
[9]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[10]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[11]

Jingzhi Yan. Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4571-4602. doi: 10.3934/dcds.2018200
[12]

Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597
[13]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[14]

Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099
[15]

C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457
[16]

Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63
[17]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881
[18]

H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361
[19]

Shigenori Matsumoto. A generic-dimensional property of the invariant measures for circle diffeomorphisms. Journal of Modern Dynamics, 2013, 7 (4) : 553-563. doi: 10.3934/jmd.2013.7.553
[20]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences