## The $$1/2$$-complex Bruno function and the Yoccoz function: a numerical study of the Marmi-Moussa-Yoccoz conjecture.(English)Zbl 1173.37324

Summary: We study the $$1/2$$-complex Bruno function and we produce an algorithm to evaluate it numerically, giving a characterization of the monoid $$\hat{\mathcal{M}}=\mathcal{M}_T\cup \mathcal{M}_S$$. We use this algorithm to test the Marmi-Moussa-Yoccoz Conjecture about the Hölder continuity of the function $$z\mapsto -i\mathbb{B}(z)+ \log U\!\left(e^{2\pi i z}\right)$$ on $$\{ z\in \mathbb{C}: \operatorname{Im} z \geq 0 \}$$, where $$\mathbb{B}$$ is the $$1/2$$-complex Bruno function and $$U$$ is the Yoccoz function. We give a positive answer to an explicit question of S. Marmi, P. Moussa and J.-C. Yoccoz [Commun. Math. Phys. 186, No. 2, 265–293 (1997; Zbl 0947.30018)].

### MSC:

 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 11A55 Continued fractions 42B25 Maximal functions, Littlewood-Paley theory 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37M99 Approximation methods and numerical treatment of dynamical systems

Zbl 0947.30018
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### References:

 [1] Berretti A., Comm. Math. Phys. 220 (4) pp 623– (2001) · Zbl 0990.37028 [2] Buff X., Experimental Mathematics 10 (4) pp 481– (2001) · Zbl 1035.37033 [3] Buff X., Invent. Math. (2003) [4] Buff X., ”The Brjuno Function Continuously Estimates the Size of Quadratic Siegel Disks.” · Zbl 1109.37040 [5] Buric N., Nonlinearity 3 pp 21– (1990) · Zbl 0734.58030 [6] Carletti T., DCDS Series A 9 (4) pp 835– (2003) · Zbl 1036.37017 [7] Carletti T., Bull. Soc. Math. France 128 pp 69– (2000) · Zbl 0997.37017 [8] Davie A. M., Nonlinearity 7 pp 219– (1994) · Zbl 0997.37500 [9] Frazier M., Littlewood-Paley Theory and the Study of Function Spaces (1991) · Zbl 0757.42006 [10] Hardy G. H., An Introduction to the Theory of Numbers, (1979) · Zbl 0423.10001 [11] Krantz S. G., Exposition. Math. 1 (3) pp 193– (1983) [12] Littlewood J. E., J. London Math. Soc. 6 pp 230– (1931) · Zbl 0002.18803 [13] de la Llave R., Experimental Math. 11 (2) pp 219– (2002) · Zbl 1116.37306 [14] Marmi S., J. Phys. A: Math. Gen. 23 pp 3447– (1990) · Zbl 0724.58037 [15] Marmi S., Communications in Mathematical Physics 186 pp 265– (1997) · Zbl 0947.30018 [16] Marmi S., Journal of AMS 14 (4) pp 783– (2001) [17] Nakada H., Keio Math. Rep. 5 pp 37– (1980) [18] Oesterle J., Asteérisque 216 pp 49– (1993) [19] Stein E. M., Singular Integrals and Differentiability Properties of Functions. (1970) · Zbl 0207.13501 [20] Yoccoz J. -C., Asteérisque 231 pp 3– (1995) [21] Yoccoz J. -C., Dynamical Systems and Small Divisors pp 125– (2002)
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