Böttcher coordinates at fixed indeterminacy points

2018 ◽  
Vol 39 (12) ◽  
pp. 3437-3456 ◽  
Author(s):  
KOHEI UENO
Keyword(s):  
Fixed Points ◽  
Skew Products ◽  
Open Set ◽  
Monomial Map ◽  
Meromorphic Maps

We first consider the dynamics of a class of meromorphic skew products having superattracting fixed points or fixed indeterminacy points at the origin. Our theorem asserts that, if a map has a suitable weight, then it is conjugate to the associated monomial map on an invariant open set whose closure contains the origin. We next extend this result to a wider class of meromorphic maps such that the eigenvalues of the associated matrices are real and greater than $1$.

scholarly journals Index theorems for meromorphic self-maps of the projective space

2014 ◽  
Author(s):  
Marco Abate
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Fixed Points ◽  
Index Theorem ◽  
Connected Components ◽  
Small Letter ◽  
Index Theorems ◽  
Dimensional Projective Space ◽  
Meromorphic Maps ◽  
Tangent To The Identity

This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More precisely, the chapter seeks to prove a particular index theorem: Let f : ℙⁿ ⇢ ℙⁿ be a meromorphic self-map of degree ν‎ + 1 ≥ 2 of the complex n-dimensional projective space. Let Σ‎(f) = Fix(f) ∪ I(f) be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let Σ‎(f) = ⊔subscript Greek Small Letter AlphaΣ‎subscript Greek Small Letter Alpha be the decomposition of Σ‎ in connected components, and denote by N the tautological line bundle of ℙⁿ. After laying out the statements under this theorem, the chapter discusses the proofs.

scholarly journals On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points II

2011 ◽  
Vol 215 (2) ◽  
pp. 177-202 ◽  
Author(s):  
Núria Fagella ◽  
Xavier Jarque ◽  
Jordi Taixés
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Meromorphic Maps

The limit set of discrete subgroups of PSL(3, ℂ)

2010 ◽  
Vol 150 (1) ◽  
pp. 129-146 ◽  
Author(s):  
WALDEMAR DEL JESÚS BARRERA VARGAS ◽  
ANGEL CANO CORDERO ◽  
JUAN PABLO NAVARRETE CARRILLO
Keyword(s):  
Fixed Points ◽  
General Position ◽  
Discrete Subgroup ◽  
Limit Set ◽  
Connected Component ◽  
Discrete Subgroups ◽  
Natural Action ◽  
Additional Hypothesis ◽  
Open Set ◽  
Complex Lines

AbstractIf Γ is a discrete subgroup of PSL(3, ℂ), it is determined the equicontinuity region Eq(Γ) of the natural action of Γ on ℙ2ℂ. It is also proved that the action restricted to Eq(Γ) is discontinuous, and Eq(Γ) agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of Γ in the sense of Kulkarni, Λ(Γ), contains at least three complex lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which Γ acts discontinuously. Moreover, if Λ(Γ) contains at least four complex lines and Γ acts on ℙ2ℂ without fixed points nor invariant complex lines, then each connected component of Eq(Γ) is a holomorphy domain and a complete Kobayashi hyperbolic space.

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

On $\alpha$-nonexpansive mapping and fixed-points in Banach spaces

2018 ◽  
Vol 2018 (-) ◽  
Author(s):  
Prondanai Kaskasem ◽  
Chakkrid Klin-eam ◽  
Suthep Suantai
Keyword(s):  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Banach Spaces

Fixed Points of Sequences of Mappings

2018 ◽  
Vol 2018 (-) ◽  
Author(s):  
C. Ganesa Moorthy ◽  
S. Iruthaya Raj
Keyword(s):  
Fixed Points
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