scholarly journals Complementary components to the cubic principal hyperbolic domain

2018 ◽  
Vol 146 (11) ◽  
pp. 4649-4660
Author(s):  
Alexander Blokh ◽  
Lex Oversteegen ◽  
Ross Ptacek ◽  
Vladlen Timorin
Keyword(s):  
Hyperbolic Domain

Hyperbolic Metric and Multiply Connected Wandering Domains of Meromorphic Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 787-810
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Analytic Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Hyperbolic Metric ◽  
Multiply Connected ◽  
The Hyperbolic Metric ◽  
Large Round ◽  
Hyperbolic Domain ◽  
Wandering Domains

AbstractIn this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.

scholarly journals The automorphism group of a certain unbounded non-hyperbolic domain

2014 ◽  
Vol 409 (2) ◽  
pp. 637-642 ◽  
Author(s):  
Hyeseon Kim ◽  
Van Thu Ninh ◽  
Atsushi Yamamori
Keyword(s):  
Automorphism Group ◽  
Hyperbolic Domain

Gravito-inertial waves in a rotating stratified sphere or spherical shell

1999 ◽  
Vol 398 ◽  
pp. 271-297 ◽  
Author(s):  
B. DINTRANS ◽  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Periodic Orbit ◽  
Spherical Shell ◽  
Periodic Orbits ◽  
Point Spectrum ◽  
Boussinesq Approximation ◽  
Diffusion Limit ◽  
Order Partial Differential Equation ◽  
Inertial Waves ◽  
Zero Diffusion Limit ◽  
Hyperbolic Domain

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [0, N], N being the Brunt–Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0, N]. It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectrum is also empty.

scholarly journals A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

2004 ◽  
Vol 206 (1) ◽  
pp. 227-252 ◽  
Author(s):  
Karen Yagdjian
Keyword(s):  
Fundamental Solution ◽  
Type Equation ◽  
Hyperbolic Domain

scholarly journals Coefficient estimates for certain subclasses of starlike functions of complex order associated with hyperbolic domain

2021 ◽  
Vol 13(62) (2) ◽  
pp. 595-610
Author(s):  
K.R. Karthikeyan ◽  
G. Murugusundaramoorthy ◽  
A. Nistor-Serban
Keyword(s):  
Analytic Functions ◽  
Starlike Functions ◽  
Coefficient Estimates ◽  
Complex Order ◽  
Coefficient Inequalities ◽  
Hyperbolic Domain

In this paper, we obtain the coefficient inequalities for functions in certain subclasses of Janowski starlike functions of complex order which are related starlike functions associated with a hyperbolic domain. Our results extend the study of various subclasses of analytic functions. Several applications of our results are also mentioned

scholarly journals The Carathéodory topology for multiply connected domains I

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mark Comerford
Keyword(s):  
Equivalent Condition ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Hyperbolic Domain ◽  
Carathéodory Topology ◽  
Hyperbolic Geodesics ◽  
Hyperbolic Domains

AbstractWe consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.

scholarly journals Hyperbolic distance versus quasihyperbolic distance in plane domains

2021 ◽  
Vol 8 (20) ◽  
pp. 578-614
Author(s):  
David Herron ◽  
Jeff Lindquist
Keyword(s):  
Euclidean Plane ◽  
Metric Spaces ◽  
The Other ◽  
Hyperbolic Distance ◽  
Other Hand ◽  
Gromov Hyperbolic ◽  
Hyperbolic Domain ◽  
Hyperbolic Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

Self-similarity of Julia sets of the composition of polynomials

1997 ◽  
Vol 17 (6) ◽  
pp. 1289-1297 ◽  
Author(s):  
MATTHIAS BÜGER
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Self Similarity ◽  
Iteration Theory ◽  
Finite Case ◽  
Self Similar ◽  
Hyperbolic Domain

In the classical iteration theory we say that for a given polynomial $f$ a point $z_0\in\C$ belongs to the Julia set if the sequence of iterates $(f^n)$ is not normal in any neighbourhood of $z_0$. In this paper, we look at the set of non-normality of $(F_n)$, $F_n:=f_n\circ\cdots\circ f_1$, where $(f_n)$ is a given sequence of polynomials of degree at least two. If we can find a hyperbolic domain $M$ which is invariant under all $f_n$, $n\in\N$, $\infty\in M$ and $F_n\to\infty\ (n\to\infty)$ locally uniformly in $M$, then we ask whether these sets of non-normality, which we will also call Julia sets, have properties which we know from the classical case. We show that the Julia set is self-similar. Furthermore, the Julia set is perfect or finite. The finite case may actually occur. We will also give some sufficient conditions for the Julia set being perfect. In the last section we give some examples of sequences of polynomials (where no domain $M$ exists) which have a pathological behaviour in contrast to the classical case.

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