scholarly journals On the Lê-Milnor fibration for real analytic maps

2016 ◽  
Vol 290 (2-3) ◽  
pp. 382-392 ◽  
Author(s):  
Aurélio Menegon Neto ◽  
José Seade
Keyword(s):  
Milnor Fibration ◽  
Real Analytic ◽  
Analytic Maps

MILNOR FIBRATIONS AND d-REGULARITY FOR REAL ANALYTIC SINGULARITIES

2010 ◽  
Vol 21 (04) ◽  
pp. 419-434 ◽  
Author(s):  
J. L. CISNEROS-MOLINA ◽  
J. SEADE ◽  
J. SNOUSSI
Keyword(s):  
Fiber Bundle ◽  
Critical Value ◽  
Singular Set ◽  
Small Sphere ◽  
Milnor Fibration ◽  
Analytic Varieties ◽  
Real Analytic ◽  
Analytic Singularities ◽  
Analytic Maps ◽  
Do So

We study Milnor fibrations of real analytic maps [Formula: see text], n ≥ p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f-1(0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom af-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/‖f‖. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in ℝn the spherefication gives a fiber bundle equivalent to the Milnor fibration.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

On the topology of real analytic maps

2014 ◽  
Vol 25 (07) ◽  
pp. 1450069 ◽  
Author(s):  
José Luis Cisneros-Molina ◽  
José Seade ◽  
Nivaldo G. Grulha
Keyword(s):  
Critical Point ◽  
Euler Characteristic ◽  
Euler Characteristics ◽  
Critical Set ◽  
Gauss Theorem ◽  
Real Analytic ◽  
The Difference ◽  
Analytic Maps ◽  
Zero Locus ◽  
Gradient Vector Field

We study the topology of the fibers of real analytic maps ℝn → ℝp, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : ℝn → ℝp, p ≥ 1, with arbitrary critical set, has a Milnor–Lê type fibration away from the discriminant. Now assume also that f has the Thom af-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : ℝn → ℝk with an isolated critical point at the origin. We have Milnor–Lê type fibrations for f and for (f, g) : ℝn → ℝp+k, and we prove for these the analogous of the classical Lê–Greuel formula, expressing the difference of the Euler characteristics of the fibers Ff and Ff,g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map [Formula: see text] on Ff associated to g; as the number of critical points of [Formula: see text] on Ff; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers Ff and Ff,g, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map.

Open-book decompositions of 𝕊5 and real singularities

2014 ◽  
Vol 25 (09) ◽  
pp. 1450085
Author(s):  
Haydée Aguilar-Cabrera
Keyword(s):  
Open Book ◽  
Isolated Singularity ◽  
Complex Singularity ◽  
Seifert Manifold ◽  
Milnor Fibration ◽  
Open Book Decompositions ◽  
The Family ◽  
Real Analytic ◽  
Almost All ◽  
Real Singularities

In this article, we study the topology of the family of real analytic germs F : (ℂ3, 0) → (ℂ, 0) given by [Formula: see text] with p, q, r ∈ ℕ, p, q, r ≥ 2 and (p, q) = 1. Such a germ has an isolated singularity at 0 and gives rise to a Milnor fibration [Formula: see text]. Moreover, it is known that the link LF is a Seifert manifold and that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of 𝕊5 given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ℂ3.

On the structure of semialgebraic sets over p-adic fields

1988 ◽  
Vol 53 (4) ◽  
pp. 1138-1164 ◽  
Author(s):  
Philip Scowcroft ◽  
Lou van den Dries
Keyword(s):  
Boolean Operations ◽  
Semialgebraic Sets ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Open Set ◽  
Real Analytic ◽  
Selection Lemma ◽  
Image Position ◽  
Analytic Maps ◽  
Curve Selection

In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:Let V ⊂ Rm be a real algebraic set, and let U ⊂ Rm be an open set defined by finitely many polynomial inequalities:Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure (U ∩ V)) then there exists a real analytic curvewith p(0) = 0 and with p(t) ∈ U ∩ V for t > 0.Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps Rn+1 → Rn. If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Qp’, we obtain a p-adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p-adic context, may be defined just as they are over the reals: namely, as those sets obtained from p-adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.

scholarly journals Uniform tail entropy for real analytic maps

2015 ◽  
Vol 183 ◽  
pp. 1-10 ◽  
Author(s):  
Gang Liao
Keyword(s):  
Real Analytic ◽  
Analytic Maps

scholarly journals Repellers for real analytic maps

1982 ◽  
Vol 2 (1) ◽  
pp. 99-107 ◽  
Author(s):  
David Ruelle
Keyword(s):  
Hausdorff Dimension ◽  
Rational Function ◽  
General Result ◽  
Julia Set ◽  
Real Analytic ◽  
Analytic Maps

AbstractThe purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see corollary 5).

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