# American Institute of Mathematical Sciences

January  2018, 38(1): 63-74. doi: 10.3934/dcds.2018003

## Surgery on Herman rings of the standard Blaschke family

 Northwest University, School of Mathematics, Xi'an Shaanxi 710127, China

Received  January 2016 Revised  August 2017 Published  September 2017

Fund Project: The author is supported by NSFC (grant No. 11426177,11301417) and NSF of Northwest University (grant No. NC14035)

Let
 $B_{\alpha ,a}$
be the Blaschke product of the following form:
 ${B_{\alpha ,a}}(z) = {e^{2\pi {\rm{\mathbf{i}}}\alpha }}{z^{d + 1}}{(\frac{{z - a}}{{1 - az}})^d}.$
If
 $B_{\alpha ,a}|_{S^1}$
is analytically linearizable, then there is a Herman ring admitting the unit circle as an invariant curve in the dynamical plane of
 $B_{\alpha ,a}$
. Given an irrational number
 $θ$
, the parameters
 $(\alpha ,a)$
such that
 $B_{\alpha ,a}|_{S^1}$
has rotation number
 $θ$
form a curve
 $T_d(θ)$
in the parameter plane. Using quasiconformal surgery, we prove that if
 $θ$
is of Brjuno type, the curve can be parameterized real analytically by the modulus of the Herman ring, from
 $a=M(θ)$
up to
 $∞$
with
 $M(θ)≥q 2d+1$
, for which the Herman ring vanishes.Moreover, we can show that for a certain set of irrational numbers
 $θ ∈ \mathcal {B}\setminus\mathcal {H}$
, the number
 $M(θ)$
is strictly greater than
 $2d+1$
and the boundary of the Herman rings consist of two quasicircles not containing any critical point.
Citation: Haifeng Chu. Surgery on Herman rings of the standard Blaschke family. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 63-74. doi: 10.3934/dcds.2018003
##### References:
 [1] L. Ahlfors, Lectures on Quasiconformal Mappings 2$^{nd}$ edition, University Lecture Series, 38 2006. doi: 10.1090/ulect/038. Google Scholar [2] V. Arnold, Small denominators I: On the mapping of a circle into itself, Nauk. Math., Series, 25 (1961), 21-96. Google Scholar [3] H. F. Chu, On the Blaschke circle diffeomorphisms, Proceedings of the American Mathematical Society, 143 (2015), 1169-1182. doi: 10.1090/S0002-9939-2014-12359-8. Google Scholar [4] N. Fagella and L. Geyer, Surgery on Herman rings of the complex standard family, Ergodic Theory and Dynamical Systems, 23 (2003), 493-508. doi: 10.1017/S0143385702001323. Google Scholar [5] L. Geyer, Siegel disks, Herman rings and Arnold family, Trans. Amer. Math. Soc., 353 (2001), 3661-3683. doi: 10.1090/S0002-9947-01-02662-9. Google Scholar [6] C. Henriksen, Holomorphic Dynamics and Herman Rings Master's thesis, Technical University of Denmark, 1997.Google Scholar [7] M. Herman, Sur les conjugaison différentiable des difféomorphismes du cercle á des rotations, Publ. Math. IHES., 49 (1979), 5-233. Google Scholar [8] M. Herman, Conjugaison quasi-symmétrique des difféomorphismes du cercle á des rotations et applications aux disques singuliers de siegel I, unpublished manuscript.Google Scholar [9] O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane Springer-Verlag, 1973. Google Scholar [10] W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, 1993. doi: 10.1007/978-3-642-78043-1. Google Scholar [11] J. Milnor, Dynamics in One Complex Variable ntroductory Lectures, 2000. doi: 10.1007/978-3-663-08092-3. Google Scholar [12] E. Risler, Linéarisation des perturbations holomorphes des rotations et applications, Mémoires de la Société Mathématique de France, 77 (1999), 1-102. Google Scholar [13] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm., 20 (1987), 1-29. doi: 10.24033/asens.1522. Google Scholar [14] J. C. Yoccoz, Analytic linearization of circle diffeomorphisms in Dynamical Systems and Small Divisors (Lecture Notes in Mathematics), Springer, Berlin, 1784 (2002), 125-173. doi: 10.1007/978-3-540-47928-4_3. Google Scholar

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##### References:
 [1] L. Ahlfors, Lectures on Quasiconformal Mappings 2$^{nd}$ edition, University Lecture Series, 38 2006. doi: 10.1090/ulect/038. Google Scholar [2] V. Arnold, Small denominators I: On the mapping of a circle into itself, Nauk. Math., Series, 25 (1961), 21-96. Google Scholar [3] H. F. Chu, On the Blaschke circle diffeomorphisms, Proceedings of the American Mathematical Society, 143 (2015), 1169-1182. doi: 10.1090/S0002-9939-2014-12359-8. Google Scholar [4] N. Fagella and L. Geyer, Surgery on Herman rings of the complex standard family, Ergodic Theory and Dynamical Systems, 23 (2003), 493-508. doi: 10.1017/S0143385702001323. Google Scholar [5] L. Geyer, Siegel disks, Herman rings and Arnold family, Trans. Amer. Math. Soc., 353 (2001), 3661-3683. doi: 10.1090/S0002-9947-01-02662-9. Google Scholar [6] C. Henriksen, Holomorphic Dynamics and Herman Rings Master's thesis, Technical University of Denmark, 1997.Google Scholar [7] M. Herman, Sur les conjugaison différentiable des difféomorphismes du cercle á des rotations, Publ. Math. IHES., 49 (1979), 5-233. Google Scholar [8] M. Herman, Conjugaison quasi-symmétrique des difféomorphismes du cercle á des rotations et applications aux disques singuliers de siegel I, unpublished manuscript.Google Scholar [9] O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane Springer-Verlag, 1973. Google Scholar [10] W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, 1993. doi: 10.1007/978-3-642-78043-1. Google Scholar [11] J. Milnor, Dynamics in One Complex Variable ntroductory Lectures, 2000. doi: 10.1007/978-3-663-08092-3. Google Scholar [12] E. Risler, Linéarisation des perturbations holomorphes des rotations et applications, Mémoires de la Société Mathématique de France, 77 (1999), 1-102. Google Scholar [13] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm., 20 (1987), 1-29. doi: 10.24033/asens.1522. Google Scholar [14] J. C. Yoccoz, Analytic linearization of circle diffeomorphisms in Dynamical Systems and Small Divisors (Lecture Notes in Mathematics), Springer, Berlin, 1784 (2002), 125-173. doi: 10.1007/978-3-540-47928-4_3. Google Scholar
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