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

Surgery on Herman rings of the standard Blaschke family

Haifeng Chu

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 & 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.  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

show all references

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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