Surgery on Herman rings of the standard Blaschke family

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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.

Keywords: Blaschke product, Herman rings, quasiconformal surgery.

Mathematics Subject Classification: Primary:37F50;Secondary:37F10.

Citation: Haifeng Chu. Surgery on Herman rings of the standard Blaschke family. Discrete and Continuous Dynamical Systems, 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.
[2]	V. Arnold, Small denominators I: On the mapping of a circle into itself, Nauk. Math., Series, 25 (1961), 21-96.
[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.
[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.
[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.
[6]	C. Henriksen, Holomorphic Dynamics and Herman Rings Master's thesis, Technical University of Denmark, 1997.
[7]	M. Herman, Sur les conjugaison différentiable des difféomorphismes du cercle á des rotations, Publ. Math. IHES., 49 (1979), 5-233.
[8]	M. Herman, Conjugaison quasi-symmétrique des difféomorphismes du cercle á des rotations et applications aux disques singuliers de siegel I, unpublished manuscript.
[9]	O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane Springer-Verlag, 1973.
[10]	W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, 1993. doi: 10.1007/978-3-642-78043-1.
[11]	J. Milnor, Dynamics in One Complex Variable ntroductory Lectures, 2000. doi: 10.1007/978-3-663-08092-3.
[12]	E. Risler, Linéarisation des perturbations holomorphes des rotations et applications, Mémoires de la Société Mathématique de France, 77 (1999), 1-102.
[13]	M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm., 20 (1987), 1-29. doi: 10.24033/asens.1522.
[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.

show all references

References:

[1]	L. Ahlfors, Lectures on Quasiconformal Mappings 2$^{nd}$ edition, University Lecture Series, 38 2006. doi: 10.1090/ulect/038.
[2]	V. Arnold, Small denominators I: On the mapping of a circle into itself, Nauk. Math., Series, 25 (1961), 21-96.
[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.
[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.
[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.
[6]	C. Henriksen, Holomorphic Dynamics and Herman Rings Master's thesis, Technical University of Denmark, 1997.
[7]	M. Herman, Sur les conjugaison différentiable des difféomorphismes du cercle á des rotations, Publ. Math. IHES., 49 (1979), 5-233.
[8]	M. Herman, Conjugaison quasi-symmétrique des difféomorphismes du cercle á des rotations et applications aux disques singuliers de siegel I, unpublished manuscript.
[9]	O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane Springer-Verlag, 1973.
[10]	W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, 1993. doi: 10.1007/978-3-642-78043-1.
[11]	J. Milnor, Dynamics in One Complex Variable ntroductory Lectures, 2000. doi: 10.1007/978-3-663-08092-3.
[12]	E. Risler, Linéarisation des perturbations holomorphes des rotations et applications, Mémoires de la Société Mathématique de France, 77 (1999), 1-102.
[13]	M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm., 20 (1987), 1-29. doi: 10.24033/asens.1522.
[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.

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