The 1/2-Complex Bruno Function and the Yoccoz Function: A Numerical Study of the Marmi-Moussa-Yoccoz Conjecture

REFERENCES

• Berretti , A. and Gentile , G. 2001 . “Bryuno Function and the Standard Map.” . Comm. Math. Phys. , 220 ( 4 ) : 623 – 656 . [Berretti and Gentile 01]
   Google Scholar
• Buff , X. , Henriksen , C. and Hubbard , J. H. 2001 . “Farey Curves.” . Experimental Mathematics , 10 ( 4 ) : 481 – 486 . [Buff et al. 01]
   Web of Science ®Google Scholar
• Buff , X. and Cheritat , A. 2003 . “Upper Bound for the Size of Quadratic Siegel Disks.” . Invent. Math. , [Buff and Cheritat 03], DOI:10.1007/s00222–003–0331–6 (pub-lished first online)
   Web of Science ®Google Scholar
• Buff , X. and Cheritat , A. “The Brjuno Function Continuously Estimates the Size of Quadratic Siegel Disks.” [Buff and Cheritat 04], Preprint, arXiv:math.DS/0401044 vl
   Google Scholar
• Buric , N. , Percival , I. C. and Vivaldi , F. 1990 . “Critical Function and Modular Smoothing.” . Nonlinearity , 3 : 21 – 37 . [Buric et al. 90]
   Web of Science ®Google Scholar
• Carletti , T. 2003 . “The Lagrange Inversion Formula on Non-Archimedian Fields. Non-Analytic Form of Differential and Finite Difference Equations.” . DCDS Series A , 9 ( 4 ) : 835 – 858 . [Carletti 03]
   Web of Science ®Google Scholar
• Carletti , T. and Marmi , S. 2000 . “Lin-earization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C, 0).” . Bull. Soc. Math. France , 128 : 69 – 85 . [Carletti and Marmi 00]
   Web of Science ®Google Scholar
• Davie , A. M. 1994 . “The Critical Function for the Semistandard Map.” . Nonlinearity , 7 : 219 – 229 . [Davie 94]
   Web of Science ®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], Conf. Board of the Math. Sciences, 79
   Google Scholar
• Hardy , G. H. and Wright , E. M. 1979 . An Introduction to the Theory of Numbers, , Fifth edition Oxford , , UK : Oxford Univ. Press. . [Hardy and Wright 79]
   Google Scholar
• Krantz , S. G. 1983 . “Lipschitz Spaces, Smoothness of Functions, and Approximation Theory.” . Exposition. Math. , 1 ( 3 ) : 193 – 260 . [Krantz 83]
   Google Scholar
• Littlewood , J. E. and Paley , R. E. A. C. 1931 . “Theorems on Fourier Series and Power Series, I.” . J. London Math. Soc. , 6 : 230 – 233 . [Littlewood and Paley 31]
   Google Scholar
• de la Llave , R. and Petrov , N. P. 2002 . “Regularity of Conjugacies between Critical Circle Maps: An Experimental Study.” . Experimental Math. , 11 ( 2 ) : 219 – 241 . [De la Llave and Petrov 02]
   Web of Science ®Google Scholar
• Marmi , S. 1990 . “Critical Functions for Complex An-alytic Maps.” . J. Phys. A: Math. Gen. , 23 : 3447 – 3474 . [Marmi 90]
   Google Scholar
• Marmi , S. , Moussa , P. and Yoccoz , J. -C. 1997 . “The Brjuno Functions and their Regularity Properties.” . Communications in Mathematical Physics , 186 : 265 – 293 . [Marmi et al. 97]
   Web of Science ®Google Scholar
• Marmi , S. , Moussa , P. and Yoccoz , J. -C. 2001 . “Complex Brjuno Functions.” . Journal of AMS , 14 ( 4 ) : 783 – 841 . [Marmi et al. 01]
   Web of Science ®Google Scholar
• Nakada , H. 1980 . “On the Invariant Measures and the Entropies for Continued Fraction Transformations.” . Keio Math. Rep. , 5 : 37 – 44 . [Nakada 80]
   Google Scholar
• Oesterle , J. 1993 . “Polylogarithmes, Seéminaire Bourbaki n. 762.” . Asteérisque , 216 : 49 – 67 . [Oesterleé93]
   Google Scholar
• Stein , E. M. 1970 . Singular Integrals and Differentiability Properties of Functions. Princeton , NJ : Princeton University Press. . [Stein 70]
   Google Scholar
• Yoccoz , J. -C. 1995 . “Theéoreéme de Siegel, polynômes quadratiques et nombres de Brjuno.” . Asteérisque , 231 : 3 – 88 . [Yoccoz 95]
   Google Scholar
• Yoccoz , J. -C. 2002 . “Analytic Linearization of Circle Diffeomorphisms.” . In Dynamical Systems and Small Divisors 125 – 174 . Berlin : Springer-Verlag. . [Yoccoz 02], Lecture Notes in Mathematics 1784
   Google Scholar