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 the geometry of the surfaces embedded in ℝ4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any generic convexly embedded 2-sphere in ℝ4 has inflection points.

The geometry of surfaces in 4-space from a contact viewpoint

Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces.

Differential Geometry from a Singularity Theory Viewpoint

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincare-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

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Inflection points and topology of surfaces in 4-space

In this article we study the singularities defined by real analytic maps 

$$ \left( {\mathbb{R}^m ,0} \right) \to \left( {\mathbb{R}^2 ,0} \right) $$

with an isolated critical point at the origin, having a Milnor fibration. It is known [14] that if such a map has rank 2 on a punctured neighbourhood of the origin, then one has a fibre bundle φ : S m−1 − → S 1, where K is the link. In this case we say that f satisfies the Milnor condition at 0 ∈ ℝ m . However, the map φ may not be the obvious map $ \frac{f} {{\parallel f\parallel }} $ as in the complex case [14, 9]. If f satisfies the Milnor condition at 0 ∈ ℝ m and for every sufficiently small sphere around the origin the map $ \frac{f} {{\parallel f\parallel }} $ defines a fibre bundle, then we say that f satisfies the strong Milnor condition at 0 ∈ ℝ m . In this article we first use well known results of various authors to translate “the Milnor condition” into a problem of finite determinacy of map germs, and we study the stability of these singularities under perturbations by higher order terms. We then complete the classification, started in [20, 21] of certain families of singularities that satisfy the (strong) Milnor condition. The simplest of these are the singularities in ℝ2 n ≅ ℂ n of the form $\{ \sum _{i = 1}^nz_i^{{a_i}}z_i^{ - {b_i}} = 0, {a_i} > {b_i} \geqslant 1\}$ We prove that these are topologically equivalent (but not analytically equivalent!) to Brieskorn-Pham singularities.

On Real Singularities with a Milnor Fibration

We prove a new Morse-Sard type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for $C^2$ mappings. We show the equivalence of three different types of regularity conditions which have been used in the literature in order to control the asymptotic behaviour of mappings. The central role of our picture is played by the $t$-regularity and its bridge toward the $\rho$-regularity which implies topological triviality at infinity.

Maria Aparecida Soares Ruas

Papers

The geometry of surfaces in 4-space from a contact viewpoint

Differential Geometry from a Singularity Theory Viewpoint

Inflection points and topology of surfaces in 4-space

On Real Singularities with a Milnor Fibration

Regularity at infinity of real mappings and a Morse–Sard theorem