A Riemann surface M is said to be K -quasiconformally homogeneous if, for every two points p, q ∈ M , there exists a K -quasiconformal homeomorphism f : M →M such that f (p) = q. In this paper, we show there exists a universal constant K > 1 such that if M is a K -quasiconformally homogeneous hyperbolic genus zero surface other than D 2 , then K ≥ K. This answers a question by Gehring and Palka [10]. Further, we show that a non-maximal hyperbolic surface of genus g ≥ 1 is not K -quasiconformally homogeneous for any finite K ≥ 1.
Abstract. Let M be a closed surface and f a diffeomorphism of M . A diffeomorphism is said to permute a dense collection of domains, if the union of the domains are dense and the iterates of any one domain are mutually disjoint. In this note, we show that if f ∈ Diff 1+α (M ), with α > 0, and permutes a dense collection of domains with bounded geometry, then f has zero topological entropy. Definitions and statement of resultsA result of Norton and Sullivan [8] states that a diffeomorphism f ∈ Diff 3 0 (T 2 ) having Denjoy-type can not have a wandering disk whose iterates have the same generic shape. By diffeomorphisms of Denjoy-type are meant diffeomorphisms of the two-torus, isotopic to the identity, that are obtained as an extension of an irrational translation of the torus, for which the semi-conjugacy has countably many non-trivial fibers. If these fibers have non-empty interior, then the corresponding diffeomorphism has a wandering disk. Further, by generic shape is meant that the only elements of SL(2, Z) preserving the shape are elements of SO(2, Z), such as round disks and squares. In a similar spirit, Bonatti, Gambaudo, Lion and Tresser in [1] show that certain infinitely renormalizable diffeomorphisms of the two-disk that are sufficiently smooth, can not have wandering domains if these domains have a certain boundedness of geometry.In this note, we study an analogous problem, namely the interplay between the geometry of iterates of domains under a diffeomorphism and its topological entropy.To state the precise result, we first need some definitions. Let (M, g) be a closed surface, that is, a smooth, closed, oriented Riemannian two-manifold, equipped with the canonical metric g induced from the standard conformal metric of the universal cover P Given f ∈ Homeo(M ), for each n ≥ 1, define the metric Ferry Kwakkel and Vladimir Markovic(n, ) separated if d n (x, y) ≥ for every x, y ∈ U with x = y. Let N (n, ) be the maximum cardinality of an (n, ) separated set. The topological entropy is defined asNext, we make precise the notion of a homeomorphism of a surface permuting a dense collection of domains. Note that we do not assume a domain to be recurrent, nor do we assume the orbit of a single domain to be dense. A wandering domain is a domain with mutually disjoint iterates under f such that the orbit of the domain is recurrent. Thus a diffeomorphism with a wandering domain with dense orbit is a special case of definition 1.1. Denote exp p : T p M → M the exponential mapping at p ∈ M . The injectivity radius at a point p ∈ M is defined as the largest radius for which exp p is a diffeomorphism. The injectivity radius ι(M ) of M is the infimum of the injectivity radii over all points where B(p, r) ⊂ M is the ball centered at p ∈ M with radius r > 0. If no such β exists, then the collection is said to have unbounded geometry.By Cl(D k ) being contractible in M we mean that Cl(D k ) is contained in an embedded topological disk in M . Our definition of bounded geometry is equivalent to the notion of bounded g...
As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.Comment: Corrected version of Fund. Math. 211 (2011), pp. 41-76, see erratum in Fund. Math. 213 (2011), p. 29
Abstract. We provide a classification of minimal sets of homeomorphisms of the twotorus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Moreover, in case (1) bounded disks are non-periodic and in case (2) all disks are non-periodic.This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.