scholarly journals Dynamics of post-critically finite maps in higher dimension

2018 ◽  
Vol 40 (2) ◽  
pp. 289-308
Author(s):  
MATTHIEU ASTORG
Keyword(s):  
Regularity Condition ◽  
Transversality Condition ◽  
Periodic Points ◽  
Higher Dimension ◽  
Critical Set ◽  
Kobayashi Hyperbolicity ◽  
Fatou Components ◽  
The Way

We study the dynamics of post-critically finite endomorphisms of $\mathbb{P}^{k}(\mathbb{C})$. We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins; thus, partially extending to any dimension is a result of Fornaess–Sibony and Rong holding in the case $k=2$.

scholarly journals Fatou maps inℙndynamics

2003 ◽  
Vol 2003 (19) ◽  
pp. 1233-1240 ◽  
Author(s):  
John W. Robertson
Keyword(s):  
Periodic Point ◽  
Complex Projective Space ◽  
Normal Family ◽  
Critical Set ◽  
Fatou Component ◽  
Hyperbolic Fixed Points ◽  
Local Stable Manifold ◽  
Fatou Components ◽  
Complex Projective ◽  
Global Stable Manifolds

We study the dynamics of a holomorphic self-mapfof complex projective space of degreed>1by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics offare a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified byf(and therefore any hyperbolic periodic point attracts a point of the critical set off). We also show that Fatou components are hyperbolically embedded inℙnand that a Fatou component which is attracted to a taut subset of itself is necessarily taut.

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Polynomial shift-like maps in ℂk

2019 ◽  
Vol 30 (12) ◽  
pp. 1950066
Author(s):  
Sayani Bera
Keyword(s):  
Periodic Points ◽  
Basins Of Attraction ◽  
Hyperbolic Polynomial ◽  
Polynomial Map ◽  
Fatou Components

The purpose of this paper is to explore a few properties of polynomial shift-like automorphisms of [Formula: see text] We first prove that a [Formula: see text]-shift-like polynomial map (say [Formula: see text]) degenerates essentially to a polynomial map in [Formula: see text]-dimensions as [Formula: see text] Second, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e. [Formula: see text], where [Formula: see text] is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

scholarly journals Equidistribution problems in complex dynamics of higher dimension

2017 ◽  
Vol 28 (07) ◽  
pp. 1750057 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Nessim Sibony
Keyword(s):  
Dynamical System ◽  
Complex Dynamics ◽  
Fundamental Problem ◽  
Arbitrary Dimension ◽  
Deep Understanding ◽  
Periodic Points ◽  
Higher Dimension ◽  
Open Problems ◽  
New Techniques ◽  
Ergodic Properties

Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented.

The Dynamics of Planar Quadratic Maps with Nonempty Bounded Critical Set

1998 ◽  
Vol 08 (01) ◽  
pp. 95-105 ◽  
Author(s):  
Chia-Hsing Nien
Keyword(s):  
Julia Sets ◽  
Periodic Points ◽  
Critical Set ◽  
Quadratic Map ◽  
Quadratic Maps

If the critical set of a planar quadratic map is bounded and nonempty, then the range is doubly covered except the image of the critical set and the region bounded by it (Fig. 1). For a map of this type, we showed that infinity is a sink and the set of bounded orbits is nonempty hence its boundary may be regarded as a generalization of Julia sets of complex quadratic maps. In this paper, we also provided evidences to conjecture that maps of this type have infinitely many periodic points.

scholarly journals Tong-il: some observations on a central problem in the psychology of religion

1975 ◽  
Vol 7 ◽  
pp. 175-188
Author(s):  
Hjalmar Sundén
Keyword(s):  
Holy Spirit ◽  
Psychology Of Religion ◽  
Central Problem ◽  
Higher Dimension ◽  
Truth Telling ◽  
The West ◽  
The World ◽  
The Holy Spirit ◽  
World Christianity ◽  
The Way

Tong-il is the Korean title of a movement known in the West as The Holy Spirit Association for the Unification of World Christianity, or the Union Church. God has formed Tong-il as an instrument of purification and renewal, bringing a new truth telling all men about the purpose of life, the responsibility of man, the way to establish a world of brotherhood and love and make the world into one family. This truth will raise Christianity to a higher dimension and give it the power and zeal which it needs to achieve God's purpose at the time of the second Advent. Tong-il works to renew Christianity, but its ultimate goal is to unite all religions, with its founder as a centre.

scholarly journals Periodic points of post-critically algebraic holomorphic endomorphisms

2021 ◽  
pp. 1-33
Author(s):  
VAN TU LE
Keyword(s):  
Complex Analysis ◽  
Complex Dynamics ◽  
Rational Maps ◽  
Higher Dimension ◽  
Holomorphic Maps ◽  
Rational Map ◽  
Critical Set ◽  
Princeton University ◽  
Dimension One ◽  
Hyperbolicity Assumption

Abstract A holomorphic endomorphism of ${{\mathbb {CP}}}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$ , a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that, when $n=2$ , the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson [Some properties of 2-critically finite holomorphic maps of P2. Ergod. Th. & Dynam. Sys.18(1) (1998), 171–187]. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one obtained by Fornæss and Sibony [Complex dynamics in higher dimension. II. Modern Methods in Complex Analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies, 137). Ed. T. Bloom, D. W. Catlin, J. P. D’Angelo and Y.-T. Siu, Princeton University Press, 1995, pp. 135–182] under a hyperbolicity assumption on the complement of the post-critical set.

NON-PERIODIC Γ-CONVERGENCE: APPLICATION TO A MAIN EXAMPLE OF A NON-FINER OSCILLATION OPERATOR IN HIGHER DIMENSION

2012 ◽  
Vol 10 (01) ◽  
pp. 67-90
Author(s):  
PABLO PEDREGAL ◽  
HELIA SERRANO
Keyword(s):  
General Setting ◽  
Divergence Form ◽  
Nonlinear Pdes ◽  
Higher Dimension ◽  
Γ Convergence ◽  
Oscillation Operator ◽  
Structural Assumption ◽  
The Way

We generalize, to a higher dimension, a main example of an operator that enjoys the NFO property introduced recently in [13]. It turns out that this issue is intimately connected to Γ-convergence of functionals in a non-periodic, general setting. Under a main structural assumption on the sequence of functionals, the Γ-convergence limit is computable. As a consequence of our analysis, we obtain the generalization just mentioned, opening thus the way to treat the existence for some nonlinear PDEs in divergence form.

Self-sacrifice for in-group's history: A diachronic perspective

2018 ◽  
Vol 41 ◽  
Author(s):  
Maria Babińska ◽  
Michal Bilewicz
Keyword(s):  
Clinical Psychology ◽  
Historical Events ◽  
Emotional Events ◽  
Reciprocal Process ◽  
The Way

AbstractThe problem of extended fusion and identification can be approached from a diachronic perspective. Based on our own research, as well as findings from the fields of social, political, and clinical psychology, we argue that the way contemporary emotional events shape local fusion is similar to the way in which historical experiences shape extended fusion. We propose a reciprocal process in which historical events shape contemporary identities, whereas contemporary identities shape interpretations of past traumas.

What is the purpose of cognition?

2020 ◽  
Vol 43 ◽  
Author(s):  
Aba Szollosi ◽  
Ben R. Newell
Keyword(s):  
Infinite Number ◽  
Human Cognition ◽  
The Way

Abstract The purpose of human cognition depends on the problem people try to solve. Defining the purpose is difficult, because people seem capable of representing problems in an infinite number of ways. The way in which the function of cognition develops needs to be central to our theories.

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