scholarly journals Fatou components with punctured limit sets

2014 ◽  
Vol 35 (5) ◽  
pp. 1380-1393 ◽  
Author(s):  
LUKA BOC-THALER ◽  
JOHN ERIK FORNÆSS ◽  
HAN PETERS
Keyword(s):  
Limit Set ◽  
Holomorphic Maps ◽  
Limit Sets ◽  
Fatou Component ◽  
Additional Hypothesis ◽  
Recurrent Case ◽  
Analytic Sets ◽  
Polynomial Endomorphisms ◽  
Fatou Components

We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^{2}$. In the recurrent case these components were classified by Fornæss and Sibony [Classification of recurrent domains for some holomorphic maps. Math. Ann. 301(4) (1995), 813–820]. Ueda [Holomorphic maps on projective spaces and continuations of Fatou maps. Michigan Math J.56(1) (2008), 145–153] completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and Peters [Classification of invariant Fatou components for dissipative Hénon maps. Preprint] classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of $\mathbb{C}^{2}$ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.

scholarly journals A non-Borel special alpha-limit set in the square

2021 ◽  
pp. 1-11
Author(s):  
STEPHEN JACKSON ◽  
BILL MANCE ◽  
SAMUEL ROTH
Keyword(s):  
Dynamical Systems ◽  
Limit Set ◽  
Limit Sets ◽  
Unit Square

Abstract We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

scholarly journals Limit Behavior of Solutions of Ordinary Linear Differential Equations

1994 ◽  
Vol 1 (3) ◽  
pp. 315-323
Author(s):  
František Neuman
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Linear Differential Equations ◽  
Limit Behavior ◽  
Limit Sets ◽  
Equivalent Linear ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

Abstract A classification of classes of equivalent linear differential equations with respect to ω-limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

Compositions of Set Operations

1970 ◽  
Vol 22 (2) ◽  
pp. 227-234
Author(s):  
D. W. Bressler ◽  
A. H. Cayford
Keyword(s):  
Topological Structure ◽  
Borel Sets ◽  
Set Operations ◽  
The Family ◽  
Analytic Sets

The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).

THE TRACE FUNCTION AND COMPLEX KLEINIAN GROUPS IN $P_{\mathbb{C}}^{2}$

2008 ◽  
Vol 19 (07) ◽  
pp. 865-890 ◽  
Author(s):  
JUAN-PABLO NAVARRETE
Keyword(s):  
Fixed Points ◽  
Kleinian Group ◽  
Cyclic Subgroup ◽  
Kleinian Groups ◽  
Trace Function ◽  
Limit Set

It is well known that the elements of PSL(2, ℂ) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in PU(2,1), then there are the equivalent notions of elliptic, parabolic or loxodromic elements; Goldman classified these transformations by their trace. In this work we extend the classification of elements of PU(2,1) to all of PSL(3, ℂ); we also extend to this setting the theorem that classifies them according to their trace. We use the notion of limit set introduced by Kulkarni, and calculate the limit set of every cyclic subgroup of PSL(3, ℂ) acting on [Formula: see text]. Given a classical Kleinian group it is possible to "suspend" this group to a subgroup of PSL(3, ℂ); we also calculate the limit set of this suspended group.

Almost sure limit sets of random samples in ℝd

1988 ◽  
Vol 20 (3) ◽  
pp. 573-599 ◽  
Author(s):  
Richard A. Davis ◽  
Edward Mulrow ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Almost Sure Convergence ◽  
Random Sets ◽  
Regularity Conditions ◽  
Limit Set ◽  
Multivariate Regular Variation ◽  
Exponential Form ◽  
Limit Sets ◽  
Random Samples ◽  
Distribution Tail

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

1995 ◽  
Vol 06 (01) ◽  
pp. 19-32 ◽  
Author(s):  
NIKOLAY GUSEVSKII ◽  
HELEN KLIMENKO
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Wild Cantor Set ◽  
Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

DYNAMICS OF MONOTONE GRAPH, DENDRITE AND DENDROID MAPS

2011 ◽  
Vol 21 (11) ◽  
pp. 3205-3215 ◽  
Author(s):  
ISSAM NAGHMOUCHI
Keyword(s):  
Periodic Point ◽  
Periodic Points ◽  
Nonempty Interior ◽  
Limit Set ◽  
Limit Sets ◽  
Dendrite Map ◽  
Graph Map ◽  
Monotone Graph

We show that, for monotone graph map f, all the ω-limit sets are finite whenever f has periodic point and for monotone dendrite map, any infinite ω-limit set does not contain periodic points. As a consequence, monotone graph and dendrite maps have no Li–Yorke pairs. However, we built a homeomorphism on a dendroid with a scrambled set having nonempty interior.

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