Regularity of Conjugacies between Critical Circle Maps: An Experimental Study

REFERENCES

• Arneodo , A. and Holschneider , M. 1988 . “Fractal dimensions and homeomorphic conjugacies,” . J. Statist. Phys. , 50 ( 5–6 ) : 995 – 1020 . [Arneodo and Holschneider 88]
   Web of Science ®Google Scholar
• Arnol'd , V. I. 1961 . “Small denominators. I. mappings of the circumference onto itself,” . Izv. Akad. Nauk SSSR Ser. Mat. , 25 : 21 – 86 . [Arnold 61], correction: 28 (1964), 479–480. English translation: Amer. Math. Soc. Transl. (2) 46 (1965), 213–284
   Google Scholar
• Berggren , L. , Borwein , J. and Borwein , P. 2000 . Pi: A Source Book, , Second Ed. New York : Springer. . [Berggren et al. 00]
   Google Scholar
• Bohr , T. , Bak , P. and Jensen , M. H. 1984 . “Transition to chaos by interaction of resonances in dissipative systems. II. Josephson Junctions, Charge-Density Waves, and Standard Maps,” . Phys. Rev. A (3) , 30 ( 4 ) : 1970 – 1981 . [Bohr et al. 84]
   Web of Science ®Google Scholar
• Borwein , J. M. and Borwein , P. B. 1984 . “The arithmetic-geometric mean and fast computation of elementary functions,” . SIAM Rev. , 26 ( 3 ) : 351 – 366 . [Borwein and Borwein 84]
   Web of Science ®Google Scholar
• Borwein , J. M. and Borwein , P. B. 1987 . Pi and the AGM: A study in analytic number theory and computational complexity New York : Wiley. . [Borwein and Borwein 87]
   Google Scholar
• Brent , R. P. 1976 . “Fast multiple-precision evaluation of elementary functions,” . J. Assoc. Comput. Mach. , 23 ( 2 ) : 242 – 251 . [Brent 76]
   Web of Science ®Google Scholar
• Briggs , K. M. The doubledouble homepage [Briggs 00], http://www.labsbt.com/people/briggsk2/doubledouble.html
   Google Scholar
• Briggs , K. M. , Dixon , T. M. and Szekeres , G. 1998 . “Analytic solutions of the Cvitanović-Feigenbaum and Feigenbaum-Kadanoff-Shenker equations,” . Intemat. J. Bifur. Chaos Appl. Sci. Engrg. , 8 ( 2 ) : 347 – 357 . [Briggs at al. 98]
   Web of Science ®Google Scholar
• Cvitanović , P. 1989 . Universality in Chaos, , Second ed. Bristol : Adam Hilger. . [Cvitanović 89]
   Google Scholar
• Daubechies , I. 1992 . Ten Lectures on Wavelets Philadelphia , PA : SIAM. . [Daubechies 92]
   Google Scholar
• Denjoy , A. 1932 . “Sur les courbes définies par les équations différentielles á la surface du tore,” . J. Math. Pures et Appl. (9. série) , 11 : 333 – 375 . [Denjoy 32]
   Google Scholar
• Dixon , T. M. , Kenny , B. G. and Briggs , K. M. 1997 . “On the universality of singular circle maps,” . Phys. Lett. A , 231 ( 5–6 ) : 359 – 366 . [Dixon et al. 97]
   Web of Science ®Google Scholar
• Delbourgo , R. and Kenny , B. G. 1991 . “Relations between universal scaling constants for the circle map near the golden mean,” . J. Math. Phys. , 32 ( 4 ) : 1045 – 1051 . [Delbourgo and Kenny 91]
   Web of Science ®Google Scholar
• Eckmann , J. P. and Epstein , H. 1986 . “On the existence of fixed points of the composition operator for circle maps,” . Comm. Math. Phys. , 107 ( 2 ) : 213 – 231 . [Eckmann and Epstein 86]
   Web of Science ®Google Scholar
• Eckmann , J.-P. and Epstein , H. “Fixed points of composition operators.” . VIIIth International Congress on Mathematical Physics . 1986 , Marseille . pp. 517 – 530 . Singapore : World Scientific. . [Eckmann and Epstein 87]
   Google Scholar
• Epstein , H. 1986 . “New proofs of the existence of the Feigenbaum functions,” . Comm. Math. Phys. , 106 ( 3 ) : 395 – 426 . [Epstein 86]
   Web of Science ®Google Scholar
• Epstein , H. 1989 . “Fixed points of composition operators. II,” . Nonlinearity , 2 ( 2 ) : 305 – 310 . [Epstein 89]
   Web of Science ®Google Scholar
• Feigenbaum , M. J. , Kadanoff , L. P. and Shenker , S. J. 1982 . “Quasiperiodicity in dissipative systems: a renormalization group analysis,” . Physica , 5D ( 2–3 ) : 370 – 386 . [Feigenbaum et al. 82]
   Google Scholar
• de Faria , E. 1999 . “Asymptotic rigidity of scaling ratios for critical circle mappings,” . Ergodic Theory Dynam. Systems , 19 ( 4 ) : 995 – 1035 . [de Faria 99]
   Google Scholar
• de Faria , E. and de Melo , W. 1999 . “Rigidity of critical circle mappings. I,” . J. Eur. Math. Soc. (JEMS) , 1 ( 4 ) : 339 – 392 . [de Faria and de Melo 99]
   Google Scholar
• de Faria , E. and de Melo , W. 2000 . “Rigidity of critical circle mappings. II,” . J. Amer. Math. Soc. , 13 ( 2 ) : 343 – 370 . [de Faria and de Melo 00]
   Web of Science ®Google Scholar
• Forsythe , G. E. , Malcolm , M. A. and Moller , C. B. 1977 . Computer Methods for Mathematical Computations Englewood Cliffs : Prentice-Hall. . [Forsythe et al. 77]
   Google Scholar
• Frazier , M. , Jawerth , B. and Weiss , G. 1991 . Littlewood-Paley Theory and the Study of Function Spaces Providence , RI : AMS. . [Frazier et al. 91]
   Google Scholar
• Ghering , F. W. and Palka , B. Quasi-Conformal Maps [Ghering and Palka, 00], Manuscript
   Google Scholar
• Glass , L. 1995 . “Cardiac arrhythmias and circle maps – a classical problem,” . Intemat. J. Bifur. Chaos Appl. Sci. Engrg. , 5 : 359 – 371 . [Glass 95]
   Google Scholar
• The GMP home page [GMP 00] http://www.swox.com/gmp/
   Google Scholar
• Greene , J. M. 1979 . “A method for determining a stochastic transition,” . J. Math. Phys. , 20 ( 6 ) : 1183 – 1201 . [Greene 79]
   Web of Science ®Google Scholar
• Hall , G. R . 1981 . “A C∞ Denjoy counterexample,” . Ergodic Theory Dynamical Systems , 1 ( 3 ) : 261 – 272 . [Hall 81]
   Google Scholar
• Hardle , W. , Kerkyacharian , G. , Picard , D. and Tsybakov , A. 1998 . Wavelets, Approximation, and Statistical Applications New York : Springer-Verlag. . [Hardle et al. 98]
   Google Scholar
• Hao , Bai-Lin . 1990 . Chaos II Singapore : World Scientific. . [Bai-Lin Hao 90]
   Google Scholar
• Herman , M. R. 1979 . “Sur la conjugaison différentiable des diffćomorphismes du cercle á des rotations,” 5 – 233 . [Herman 79], Inst. Hautes Études Sci. Publ. Math. No. 49
   Google Scholar
• Herman , M. R. “Sur les difféomorphismes du cercle de nombre de rotation de type constant.” . Conference on Harmonic Analysis in Honor of Antoni Zygmund . 1981 , Chicago , Ill. . pp. 708 – 725 . Belmont , CA : Wadsworth. . [Herman 83]
   Google Scholar
• Hernández , E. and Weiss , G. 1996 . A First Course on Wavelets Boca Raton , FL : CRC Press. . [Hernández and Weiss 96]
   Google Scholar
• Holschneider , M. and Tchamitchian , Ph. 1991 . “Pointwise analysis of Riemann's “nondifferentiable” function,” . Invent. Math. , 105 : 157 – 175 . [Holschneider and Tchamitchian 91]
   Web of Science ®Google Scholar
• Hu , B. , Valinia , A. and Piro , O. 1990 . “Universality and asymptotic limits of the scaling exponents in circle maps,” . Phys. Lett. A , 144 ( 1 ) : 7 – 10 . [Hu et al. 90]
   Web of Science ®Google Scholar
• Hu , J. and Sullivan , D. P. 1997 . “Topological conjugacy of circle diffeomorphisms,” . Ergodic Theory Dynam. Systems , 17 ( 1 ) : 173 – 186 . [Hu and Sullivan 97]
   Google Scholar
• Ignatov , A. M. 1995 . “Trivelpiece-Gould modes in a corrugated plasma slab,” . Phys. Rev. E , 51 ( 2 ) : 1391 – 1399 . [Ignatov 95]
   Web of Science ®Google Scholar
• Jaffard , S. 1997 . “Multifractal formalism for functions. I. Results valid for all functions. II. Self-similar functions,” . SIAM J. Math. Anal. , 28 ( 4 ) : 944 – 970 . [Jaffard 97], 971–998
   Web of Science ®Google Scholar
• Jaffard , S. and Meyer , Y. 1996 . “Wavelet methods for pointwise regularity and local oscillations of functions,” . Mem. Amer. Math. Soc. , 123 ( no. 587 ) [Jaffard and Meyer 96]
   Web of Science ®Google Scholar
• Katok , A. and Hasselblatt , B. 1995 . Introduction to the Modem Theory of Dynamical Systems Cambridge : Cambridge University Press. . [Katok and Hasselblatt 95]
   Google Scholar
• Katznelson , Y. and Ornstein , D. 1989 . “The differentiability of the conjugation of certain diffeomorphisms of the circle,” . Ergodic Theory Dynamical Systems , 9 ( 4 ) : 643 – 680 . [Katznelson and Ornstein 89]
   Google Scholar
• Kim , S. and Ostlund , S. 1989 . “Universal scaling in circle maps,” . Physica , 39D ( 2–3 ) : 365 – 392 . [Kim and Ostlund 89]
   Google Scholar
• Krantz , S. G. 1983 . “Lipschitz spaces, smoothness of functions, and approximation theory,” . Exposition. Math. , 1 ( 3 ) : 193 – 260 . [Krantz 83]
   Google Scholar
• Lanford , O. E. III . 1986 . “Renormalization group methods for circle mappings.”. ” . In Statistical Mechanics and Field Theory: Mathematical Aspects (Groningen, 1985) 176 – 189 . Berlin : Springer. . [Lanford 86], Lecture Notes in Phys., 257
   Google Scholar
• Lanford , O. E. III . “Renormalization group methods for critical circle mappings with general rotation number.” . VIIIth International Congress on Mathematical Physics . 1986 , Marseille . pp. 532 – 536 . Singapore : World Scientific. . [Lanford 87]
   Google Scholar
• Lanford , O. E. III . 1988 . “Renormalization group methods for circle mappings.”. ” . In Nonlinear Evolution and Chaotic Phenomena (Noto, 1987) 25 – 36 . New York : Plenum. . [Lanford 88], NATO Adv. Sci. Inst. Ser. B Phys., 176
   Google Scholar
• Lanford , O. E. III and de la Llave , R. 1984 . “Solution of the functional equation for critical circle mappings with golden rotation number.” [Lanford and de la Llave 84], Manuscript
   Google Scholar
• de la Llave , R. and Petrov , N. 1999 . “Theory of circle maps and the problem of onedimensional optical resonator with a periodically moving wall,” . Phys. Rev. E (3) , 59 ( 6 ) : 6637 – 6651 . [de la Llave and Petrov 99]
   Web of Science ®Google Scholar
• de la Llave , R. and Schafer , R. P. 1996 . “Rigidity properties of one dimensional expanding maps and applications to renormalization.” [de la Llave and Schafer, 96], Manuscript
   Google Scholar
• Louis , A. K. , Maaß , P. and Rieder , A. 1997 . Wavelets: Theory and Applications Chichester : Wiley. . [Louis et al. 97]
   Google Scholar
• Mallat , S. 1998 . A Wavelet Tour of Signal Processing San Diego , CA : Academic Press. . [Mallat 98]
   Google Scholar
• de Melo , W. 1998 . “Rigidity and renormalization in one-dimensional dynamical systems.” . Proceedings of the International Congress of Mathematicians . 1998 , Berlin . Vol. II , pp. 765 – 778 . [de Melo 98], Doc. Math. 1998, Extra Vol. II (electronic)
   Google Scholar
• de Melo , W. and Van Strien , S. 1993 . One-Dimensional Dynamics Berlin : Springer. . [de Melo and Van Strien 98]
   Google Scholar
• Mestel , B. 1984 . [Mestel 84], Univ. of Warwick thesis
   Google Scholar
• Meyer , Y. 1998 . Wavelets, Vibrations and Scalings Providence , RI : AMS. . [Meyer 98]
   Google Scholar
• Meyer , Y. 1990 . Ondelettes et Opérateurs. I Paris : Hermann. . [Meyer 90]
   Google Scholar
• Moser , J. 1966 . “A rapid convergent iteration method and non-linear differential equations. II.” . Ann. Scuola Norm. Sup. Pisa (3) , 20 : 499 – 535 . [Moser 66]
   Google Scholar
• Moser , J. and Pöschel , J. 1984 . “An extension of a result by Dinaburg and Sinai on quasiperiodic potentials,” . Comment. Math. Helv. , 59 ( 1 ) : 39 – 85 . [Moser and Pöschel 84]
   Web of Science ®Google Scholar
• Ostlund , S. , Rand , D. , Sethna , J. and Siggia , E. 1983 . “Universal properties of the transition from quasiperiodicity to chaos in dissipative systems,” . Physica , 8D ( 3 ) : 303 – 342 . [Ostlund et al. 83]
   Google Scholar
• Pinto , A. A. and Rand , D. A. 1992 . “Global phase space universality, smooth conjugacies and renormalization. II. The Cκ+α case using rapid convergence of Markov families,” . Nonlinearity , 5 ( 1 ) : 49 – 79 . [Pinto and Rand 92]
   Web of Science ®Google Scholar
• Poincaré , H. 1885 . “Sur le courbes définies par les equations différentielles,” . J. Math Pures et Appl. (4. série) , 1 : 167 – 244 . [Poincaré 1885], Reprinted in (Euvres de Henri Poincaré, Tome I, Gauthier-Villars, Paris, 1928
   Google Scholar
• Pommeau , Y. and Manneville , P. 1980 . “Intermittent transition to turbulence in dissipative dynamical systems,” . Comm. Math. Phys. , 74 ( 2 ) : 189 – 197 . [Pommeau and Manneville 80]
   Web of Science ®Google Scholar
• Press , W. H. , Teukolsky , S. A. , Vetterling , W. T. and Flannery , B. P. 1992 . Numerical Recipes in C, , Second ed. Cambridge : Cambridge University Press. . [Press et al. 92], Also available at http://www.nr.com/
   Google Scholar
• Rand , D. A. 1987 . “Fractal bifurcation sets, renormalization strange sets and their universal invariants,” . Proc. R. Soc. Lond. , A 413 : 45 – 61 . [Rand 87]
   Web of Science ®Google Scholar
• Rand , D. A. 1988 . “Global phase space universality, smooth conjugacies and renormalisation. I. The C1+α case,” . Nonlinearity , 1 ( 1 ) : 181 – 202 . [Rand 88a]
   Web of Science ®Google Scholar
• Rand , D. 1988 . “Universality and renormalisation in dynamical systems.”. ” . In New Directions in Dynamical Systems 1 – 56 . Cambridge : Cambridge Univ. Press. . [Rand 88b]
   Google Scholar
• Rand , D. A. 1992 . “Existence, nonexistence and universal breakdown of dissipative golden invariant tori. I. Golden critical circle maps,” . Nonlinearity , 5 ( 3 ) : 639 – 662 . [Rand 92]
   Web of Science ®Google Scholar
• Rand , D. , Ostlund , S. , Sethna , J. and Siggia , E. D. 1982 . “Universal transition from quasiperiodicity to chaos in dissipative systems,” . Phys. Rev. Lett. , 49 ( 2 ) : 132 – 135 . [Rand et al. 82]
   Web of Science ®Google Scholar
• Shenker , S. 1982 . “Scaling behavior in a map of a circle onto itself: empirical results,” . Physica , 5D ( 2–3 ) : 405 – 411 . [Shenker 82]
   Google Scholar
• Shraiman , B. I. 1984 . “Transition from quasiperiodicity to chaos: a perturbative renormalization-group approach,” . Phys. Rev. A (3) , 29 ( 6 ) : 3464 – 3466 . [Shraiman 84]
   Web of Science ®Google Scholar
• Sinai , Ya. G. and Khanin , K. M. 1989 . “Smoothness of conjugacies of diffeomorphisms of the circle with rotations,” . Uspekhi Mat. Nauk , 44 ( 1(265) : 57 – 82 . [Sinai and Khanin 89], 247. English translation: Russian Math. Surveys 44:1 (1989), 69–99
   Google Scholar
• Stein , E. 1970 . Singular Integrals and Differentiability Properties of Functions Princeton : Princeton University Press. . [Stein 70]
   Google Scholar
• Światek , G. 1991 . “Circle homeomorphisms with flat critical points,” . Fund. Math. , 138 ( 3 ) : 205 – 217 . [Światek 91]
   Web of Science ®Google Scholar
• Światek , G. 1998 . “On critical circle homeomorphisms,” . Bol. Soc. Brasil. Mat. (N.S.) , 29 ( 2 ) : 329 – 351 . [Światek 98]
   Google Scholar
• Tangerman , F. M. and Veerman , J. J. P. 1991 . “Scalings in circle maps. II,” . Comm. Math. Phys. , 141 ( 2 ) : 279 – 291 . [Tangerman and Veerman 91]
   Web of Science ®Google Scholar
• Väisälä , J. 1971 . Lectures on n-Dimensional Quasicon-formal Mappings (Lecture Notes in Mathematics, Vol. 229) Berlin : Springer. . [Väisälä 71]
   Google Scholar
• Veerman , J. J. P. and Tangerman , F. M. 1990 . “Scalings in circle maps. I,” . Comm. Math. Phys. , 134 ( 1 ) : 89 – 107 . [Veerman and Tangerman 90]
   Web of Science ®Google Scholar
• Wickerhauser , M. V. 1994 . Adapted Wavelet Analysis from Theory to Software Wellesley , MA : A K Peters. . [Wickerhauser 94], Software available at http://www.math.vrustl.edu/~victor/
   Google Scholar
• Yampolsky , M. 1999 . “Complex bounds for renormalization of critical circle maps,” . Ergodic Theory Dynamical Systems , 19 ( 1 ) : 227 – 257 . [Yampolsky 99]
   Google Scholar
• Yoccoz , J.-C. 1984 . “II n'y a pas de contre-exemple de Denjoy analytique,” . C. R. Acad Sc. Paris Sér. I Math. , 298 ( 7 ) : 141 – 144 . [Yoccoz 84a]
   Google Scholar
• Yoccoz , J.-C. 1984 . “Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,” . Ann. Sci. École Norm. Sup. (4) , 17 ( 3 ) : 333 – 359 . [Yoccoz 84b]
   Web of Science ®Google Scholar