https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincare ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

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Singularities of hyperbolic Gauss maps

Mixed functions are analytic functions in variables z1, ..., zn and their conjugates $\bar z_1$, ..., $\bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

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Non-degenerate mixed functions

Group extensions of p-adic and adelic linear groups . On the differential equations satisfied by period matrices . Théorème de lefschetz et critères de dégénérescence de suites spectrales . Stability of C mappings, III : finitely determined map-germs

We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres, answering a question proposed by Cecil and Ryan. For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that horo-tight spheres in hyperbolic space are metric spheres. We also prove that horo-tight hyperspheres are characterized by the property that both of its total absolute horospherical curvatures attend their minimum value. We also introduce the notion of weak horo-tightness: an immersion is weak horo-tight if only one of its total absolute curvature attends its minimum. We prove a characterization theorem for weak horo-tight hyperspheres.

Horo-tight spheres in hyperbolic space

We study some properties of submanifolds in Euclidean space concern- ing their generic contacts with straight lines and hyperplanes and expose some duality relation associated to these contacts.

/pdf/singularities-and-duality-in-the-flat-geometry-of-1xjkibz5ue.pdf

Singularities and Duality in the Flat Geometry of Submanifolds of Euclidean Spaces

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

Generalized distance-squared mappings of the plane into the plane

The constancy of the Milnor number has several characterizations which were summarized by Greuel in 1986. This paper presents a study of these characterizations in the case of families of functions with isolated singularities defined on an analytic variety.

Invariants of Topological Relative Right Equivalences

We fix a system of local coordinates x of C. Consider the ring On of holomorphic germs f : (C, 0) → C and denote by mn its maximal ideal. Due to identification between On and the ring of convergent power series C{x1, . . . , xn} we identify a germ f ∈ On with its power series f(x) = ∑ aαx , where x = x1 1 . . . x αn n . The Milnor number of a germ f with isolated singularity, denoted by μ(f), is algebraically defined as the dimC On/Jf , where Jf denotes the ideal generated by partial derivatives ∂f/∂x1, . . ., ∂f/∂xn and the multiplicity m(f) is the lowest degree in the power series expansion of f at 0 ∈ C. A deformation F : (C×C, 0) → (C, 0), F (x, t) = ft(x), of f is μ-constant if μ(ft) = μ(f) for small values of t. We denote by JF = ⟨∂F/∂x1, . . . , ∂F/∂xn⟩, the ideal in On+1 generated by the partial derivatives of F with respect to the variables x1, . . . , xn. The Milnor number is a topological invariant of the singularity, more precisely if two germs of complex hypersurfaces with isolated singularities are homeomorphic, then have the same Milnor number. We also have by [9], for n = 3, that if a family of hypersurfaces

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004113000297

Maria Aparecida Soares Ruas

Papers

Horo-tight spheres in hyperbolic space

Singularities and Duality in the Flat Geometry of Submanifolds of Euclidean Spaces

Generalized distance-squared mappings of the plane into the plane

Invariants of Topological Relative Right Equivalences

Invariants of topological relative right equivalences