Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T16:31:29.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Loewner evolution of hedgehogs and 2-conformal measures of circle maps

Part of: Low-dimensional dynamical systems Complex dynamical systems

Published online by Cambridge University Press:  28 September 2020

KINGSHOOK BISWAS
KINGSHOOK BISWAS*
Affiliation:
Ramakrishna Mission Vivekananda University, Belur Math, WB-711202, India (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in ${\mathbb C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $(g_t)$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $(g_t)$ is the orbit of a locally defined semigroup $(\Phi _t)$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $(K_t)$ . We show that the Loewner measures $(\mu _t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(g_t)$ .

Keywords

Loewner evolutionconformal measuresindifferent fixed pointscircle maps

MSC classification

Primary: 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Secondary: 37E10: Maps of the circle
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2734 - 2753
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Biswas, K.. Smooth combs inside hedgehogs. Discrete Contin. Dyn. Syst. 12(5) (2005), 853–880. CrossRefGoogle Scholar
Biswas, K.. Hedgehogs of Hausdorff dimension one. Ergod. Th. & Dynam. Sys. 28(6) (2008), 1713–1727.CrossRefGoogle Scholar
Biswas, K.. Positive area and inaccessible fixed points for hedgehogs. Ergod. Th. & Dynam. Sys. 36(6) (2016), 1839–1850.CrossRefGoogle Scholar
Biswas, K. and Pérez-Marco, R.. Uniformization of higher genus finite type log-Riemann surfaces. Preprint, 2013, http://arxiv.org/pdf/1305.2339.pdf.Google Scholar
Biswas, K. and Pérez-Marco, R.. Caratheodory convergence of log-Riemann surfaces and Euler’s formula. Contemp. Math. 639 (2015), 197–203.CrossRefGoogle Scholar
Biswas, K. and Pérez-Marco, R.. Uniformization of simply connected finite type log-Riemann surfaces. Contemp. Math. 639 (2015), 205–216.CrossRefGoogle Scholar
Brjuno, A. D.. Analytical form of differential equations. Trans. Moscow Math. Soc. 25 (1971), 131–288.Google Scholar
Chéritat, A.. Relatively compact Siegel disks with non-locally connected boundaries. Math. Ann. 349(3) (2011), 529–542.CrossRefGoogle Scholar
Lanza de Cristoforis, M. and Preciso, L.. Differentiability properties of some nonlinear operators associated to the conformal welding in Schauder spaces. Hiroshima Math. J. 33 (2003), 59–86.CrossRefGoogle Scholar
Lanza de Cristoforis, M. and Rogosin, S. V.. Analyticity of a nonlinear operator associated to the conformal representation in Schauder spaces. An integral equation approach. Math. Nachr. 220 (2000), 59–77.3.0.CO;2-7>CrossRefGoogle Scholar
Douady, R. and Yoccoz, J. C.. Nombre de rotation des diffeomorphismes du cercle et mesures automorphes. Regul. Chaotic Dyn. 4(4) (1999), 3–24.CrossRefGoogle Scholar
Pérez-Marco, R.. Sur les dynamiques holomorphes non-linearisables et une conjecture de V.I. Arnold. Ann. Sci. Éc. Norm. Supér. 26 (1993), 565–644.CrossRefGoogle Scholar
Pérez-Marco, R.. Topology of Julia sets and hedgehogs. Preprint, Université de Paris-Sud, 1994.Google Scholar
Pérez-Marco, R.. Uncountable number of symmetries for non-linearisable holomorphic dynamics. Invent. Math. 119 (1995), 67–127.CrossRefGoogle Scholar
Pérez-Marco, R.. Hedgehogs dynamics. Preprint, University of California, Los Angeles, 1996.Google Scholar
Pérez-Marco, R.. Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294.CrossRefGoogle Scholar
Pérez-Marco, R.. Siegel disks with smooth boundary. Preprint, 2000.Google Scholar
Poltoratski, A. G.. On the distribution of boundary values of Cauchy integrals. Proc. Amer. Math. Soc. 124(8) (1996), 2455–2463.CrossRefGoogle Scholar
Pommerenke, C.. Univalent Functions. Vandenhoeck and Ruprecht, Göttingen, 1975.Google Scholar
Siegel, C. L.. Iteration of analytic functions. Ann. Math. 43 (1942), 807–812.CrossRefGoogle Scholar
Yoccoz, J. C.. Petits diviseurs en dimension 1. Astérisque 231(3) (1995), 242 pp.Google Scholar