Distortion and Distribution of Sets Under Inner Functions
- PDF / 300,976 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 15 Downloads / 197 Views
Distortion and Distribution of Sets Under Inner Functions Matteo Levi1 · Artur Nicolau2
· Odí Soler i Gibert2
Received: 16 January 2019 © Mathematica Josephina, Inc. 2019
Abstract It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied by Fernández and Pestana. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. We use our results to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half-plane. Keywords Inner functions · Boundary fixed points · Angular derivatives · Hausdorff contents Mathematics Subject Classification 30J05 · 30H05
1 Introduction Let D be the open unit disc of the complex plane. An analytic mapping f : D → D is called inner if |limr →1 f (r ξ )| = 1 for almost every point (a.e.) ξ of the unit circle ∂D. Matteo Levi is partially supported by the 2015 PRIN Grant Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis of the Italian Ministry of Education (MIUR). Matteo Levi, Artur Nicolau, and Odí Soler i Gibert are supported in part by the Generalitat de Catalunya (Grant 2017 SGR 395) and the Spanish Ministerio de Ciencia e Innovación (Projects MTM2014-51824-P, MTM2017-85666-P).
B
Artur Nicolau [email protected] Matteo Levi [email protected] Odí Soler i Gibert [email protected]
1
Dipartimento di Matematica, Università di Bologna, Via Zamboni 33, 40126 Bologna, Italy
2
Departament de Matemàtiques, Universitat Autònoma De Barcelona, Edifici C, 08193 Bellaterra, Catalunya, Spain
123
M. Levi et al.
Hence, an inner function f induces a map defined at almost every point ξ ∈ ∂D by f ∗ (ξ ) = limr →1 f (r ξ ), which we will denote by f as well. This induced map lacks the regularity of the inner function itself and it is actually discontinuous at every point ξ ∈ ∂D where f does not extend analytically. More concretely, for any fixed ξ ∈ ∂D where f does not extend analytically and any η ∈ ∂D there exists a sequence ξn → ξ such that f (ξn ) → η (see [5, p. 77], and [7, p. 4]). We are interested in studying certain invariance and distortion properties of measures and Hausdorff contents of sets in the unit circle under the action of inner functions. Let f : D → D be an analytic mapping. We say that a point p ∈ ∂D is a boundary Fatou point of f if f ( p) = limr →1 f (r p) exists and f ( p) ∈ ∂D. Hence, the set of boundary Fatou points of an inner function has full measure. For 0 < β < 1 and p ∈ ∂D, let β ( p) = {z ∈ D : |z − p| < β(1 − |z|)} be the Stolz angle with opening β and vertex at p. A holomorphic self-map f of the unit disc has finite angular derivative at p ∈ ∂D if there is a point η ∈ ∂D and β > 0 such t
Data Loading...
