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Simultaneous Uniformization for Uniformly Quasisymmetric Circle Dynamical Systems

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Simultaneous Uniformization for Uniformly Quasisymmetric Circle Dynamical Systems Dynamical Systems An International Journal ISSN: 1468-9367 (Print) 1468-9375 (Online) Journal homepage: http://www.tandfonline.com/loi/cdss20 Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems Frederick P. Gardiner & Yunping Jiang To cite this article: Frederick P. Gardiner & Yunping Jiang (2015) Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems, Dynamical Systems, 30:4, 543-565, DOI: 10.1080/14689367.2015.1082537 To link to this article: http://dx.doi.org/10.1080/14689367.2015.1082537 Published online: 18 Sep 2015. Submit your article to this journal Article views: 69 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=cdss20 Download by: [Massey University Library] Date: 18 April 2016, At: 15:56 Dynamical Systems, 2015 Vol. 30, No. 4, 543–565, http://dx.doi.org/10.1080/14689367.2015.1082537 Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems Frederick P. Gardinera and Yunping Jianga,b* aDepartment of Mathematics, Graduate School of the City University of New York, 365 Fifth Avenue, New York, NY 10016, USA; bDepartment of Mathematics, Queens College of the City University of New York, Flushing, NY 11367-1597, USA (Received 5 January 2015; final version received 10 August 2015) In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisym- metric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures. Keywords: simultaneous uniformization; uniformly quasisymmetric circle endomor- phism; uniformly symmetric circle endomorphism; normalized quasidisc pairing; con- formal welding; dynamical welding 2010 Mathematics Subject Classification: Primary 37D35, 37F30; Secondary 37E10, 37A05 1. Introduction In [1] and [2] Ahlfors establishes a one-to-one correspondence between a class QC of nor- malized sets in the Riemann sphere with a class QS of normalized self-homeomorphisms of the unit circle. The sets (called normalized quasicircles) in QC are those that can be Downloaded by [Massey University Library] at 15:56 18 April 2016 realized as the image of the unit circle by a K-quasiconformal self-homeomorphism H of the Riemann sphere C that fixes the points 0, 1 and ∞ for some value of K > 1. The home- omorphisms (called quasisymmetric maps) in QS are those that are orientation preserving, that fix the point 1, and that satisfy inequality (1) in Definition 1. This correspondence is now called conformal welding. In this paper we use the same idea to weld together qua- sisymmetrically equivalent uniformly quasisymmetric circle endomorphisms of the same degree, (see Definition 2). Each element in QC determines a normalized quasidisc pairing A = (A0,∂A,A∞), *Corresponding author. Email: [email protected] C 2015 Taylor & Francis 544 F.P.Gardiner and Y. Jiang where ∂A is the set, A0 is the component of C \ ∂Athat contains 0 and A∞ is the component of C \ ∂A that contains ∞. We will prove that every uniformly quasisymmetric circle endomorphism fa corresponds uniquely to a quasidisc pairing A (Theorem 8). Given any two uniformly quasisymmetric circle endomorphisms fa and fb of degree d ≥ 2 and their associated quasidisc pairings A and B, we find another quasidisc pairing AB and a unique branched covering FAB of the Riemann sphere C preserving ∂AB and having only two local degree d fixed branched points at 0 and ∞ such that FAB has the same dynamics as fa when it is viewed as a map on the boundary of AB0 and FAB has the same dynamics as fb when it is viewed as a map on the boundary of AB∞. Note that ∂AB is the common boundary of AB0 and AB∞. We call this configuration together with expanding map FAB the simultaneous uniformization, (see Theorem 9). Given this normalized quasidisc pairing AB for the simultaneous uniformization of fa and fb,weletz0 be the Riemann mapping that fixes the points 0 and 1 and maps (AB)0 onto the unit disc = {z:|z| < 1} and, similarly, we let z∞ be the Riemann mapping that fixes the points 1 and ∞ and maps (AB)∞ onto the unit disc C \ ={z : |z| > 1}. Then the horizontal trajectories of the double pole quadratic differentials 2 2 dz0 dz∞ q0 = and q∞ = z0 z∞ fill in Riemann sphere (see [3]). Each horizontal trajectory of q0 meets a unique such horizontal trajectory of q∞ on the quasicircle ∂AB. This picture yields a transformation TAB that transforms any probability measure on the tangent circle at 0 to the Riemann sphere to a probability measure on the tangent circle at ∞. It has the cocycle property, TAC = TBC ◦ TAB . The construction of TAB enables us to study the correspondence between the space of all probability invariant measures for fa and the space of all probability invariant measures for fb by using the space of all probability invariant measures for fAB on ∂AB. We organize this paper as follows. In Section 2, we give the definition of a quasisymmet- ric homeomorphism and the definition of a uniformly quasisymmetric circle endomorphism. In Section 3, we review the measurable Riemann mapping theorem. In Section 4, we define a normalized quasidisc pairing and give the lower sided conformal welding and the upper Downloaded by [Massey University Library] at 15:56 18 April 2016 sided conformal welding in Theorem 5. In Section 5, we study the one-sided conformal fac- torization for a quasisymmetric homeomorphism from a quasicircle to another quasicircle, which we call concatenation, in Theorem 6. In Section 6, we study the conformal welding and one-sided conformal realization for a uniformly quasisymmetric circle dynamical sys- tem in Theorem 8. In the same section, we state our simultaneous uniformization for two uniformly quasisymmetric circle dynamical systems in Theorem 9. Since the proof is the same as the proof of Theorem 11, we delay the proof to the next section. Section 7 explains simultaneous uniformization for two uniformly symmetric circle dynamical systems. In this section we give a detailed proof of Theorem 11 which also serves as the proof of The- orem 9. In Section 8, we define the transformation TW associated with a quasidisc pairing W. This transformation is a self-map of the space of all probability measures on the unit circle. We show that this transformation satisfies the cocycle property. In Section 9,weuse the simultaneous uniformization and the transformation to formulate several conjectures equivalent to the uniqueness conjecture for the invariant symmetric probability measure. In Dynamical Systems 545 a subsequent paper, we will use the tools developed in this paper to study these equivalent conjectures. 2. Quasisymmetric circle homeomorphisms and uniformly quasisymmetric circle endomorphisms Let T ={z = e2πiθ ∈ C ;0≤ θ<1} be the unit circle and h be an orientation preserving self-homeomorphism of T. Suppose also that f : T → T is an orientation preserving unramified covering map of degree d ≥ 2. We call such a mapping a circle endomorphism. For an interval I in T, we use the notation | I | to denote its arc length. We say two contiguous intervals L and R in T are symmetric if |L| = |R|. Since for any t > 0thesetof symmetric contiguous pairs of intervals L and R in T with |L| = |R| ≥ t forms a compact set, the continuous function |h(R)| log |h(L)| has positive minimum. Thus, there is a number Mt ≥ 1 such that for all symmetric contiguous intervals L and R with |L| = |R| ≥ t, |h(L)| 1/Mt ≤ ≤ Mt . |h(R)| In general the constant Mt depends on h and t. Definition 1 (Quasisymmetric and symmetric circle homeomorphism): We call the circle homeomorphism h quasisymmetric if there is a number M (not depending on t) such that Mt ≤ M for all t > 0. Equivalently, h is quasisymmetric if there is a constant M > 0 such that 1 |h(L)| ≤ ≤ M (1) M |h(R)| Downloaded by [Massey University Library] at 15:56 18 April 2016 for all symmetric contiguous intervals L and R. Furthermore, if we can choose Mt ≥ 1 such + + that Mt → 1 as t → 0 , then we call h a symmetric homeomorphism. Assume L and R are in T and I = L ∪ R is an interval properly contained in T. Then for each n > 0, f−n(I) has dn connected components n In,i = Ln,i ∪ Rn,i , 1 ≤ i ≤ d . n n Then f : In, i → I is a homeomorphism for each 1 ≤ i ≤ d and n n f (Ln,i ) = L and f (Rn,i ) = R. 546 F.P.Gardiner and Y. Jiang Definition 2 (Uniformly quasisymmetric and symmetric circle endomorphism): The circle endomorphism f is called uniformly quasisymmetric if there is a number M ≥ 1 (not depending on t) such that 1 |Ln,i | ≤ ≤ M (2) M |Rn,i | for all symmetric contiguous intervals L and R with |L| = |R| = t,alln > 0, and all 1 ≤ i ≤ dn. Furthermore, we call the circle endomorphism f uniformly symmetric if we can find + + Mt ≥ 1 such that Mt → 1 as t → 0 such that 1 |Ln,i | ≤ ≤ Mt (3) Mt |Rn,i | for all symmetric contiguous intervals L and R with |L| = |R| = t,alln > 0, and all 1 ≤ i ≤ dn.
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