On the Dynamics of nnn-Dimensional Quadratic Endomorphisms (original) (raw)

Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points

Nonlinear Analysis: Theory, Methods & Applications, 1994

The basin of attraction of a set A (invariant or mapped into itself by t), denoted by a)(A), is the set (open) of all the points x such that t"(x) + A as n + 00. A closed invariant set a is an attracting set if a neighbourhood U of C? exists such that t(U) c U and t"(x) + C? as n-+ 00 for any x E LT. We say that an interval Z is absorbing if: (i) its boundary consists of critical points (of some rank) of t; (ii) t(1) 5 I; and (iii) Z is strictly included in its basin of attraction a>(Z). An absorbing interval Z is said to be proper if both: (j) there exists at most one invariant attracting set Q. included in I; and (jj) a = nkkO t"(Z).

On the invariant sets of a family of quadratic maps

Communications in Mathematical Physics, 1983

The Julia set B λ for the mapping z->(z-λ) 2 is considered, where λ is a complex parameter. For λ ^2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure on B λ are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the threeterm recurrence relations are given. A unifying theme is the relationship between B λ and /ί-chains λ± ]/(λ± ]/{λ± ...), which is explored for-\^λ^2 and for λe(£ with \λ\^4, with the aid of the Bδttcher equation. Then B ; is shown to be a Holder continuous curve for |A|<^.

Some generic properties of partially hyperbolic endomorphisms*

Nonlinearity

In this work, we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class of partially hyperbolic endomorphism C 1-sufficiently close to a hyperbolic linear endomorphism. Indeed such obstructions are related to the number of center directions of a point. We provide examples illustrating these obstructions. We show that for a manifold M with dimension n ⩾ 3, admitting a non-invertible partially hyperbolic endomorphisms, there is a C 1 open and dense subset U of all partially hyperbolic endomorphisms with degree d ⩾ n, such that any f ∈ U is neither c nor u special.

Invariant measures for typical quadratic maps

Astérisque, 2000

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.

Convexity of a small ball under quadratic map

Linear Algebra and its Applications, 2016

We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak [1] in the case of quadratic map. We also generalize the notion of the joint numerical range of m-tuple of matrices by adding vector-dependent inhomogeneous terms and provide a sufficient condition for its convexity.

Quadratic volume preserving maps: an extension of a result of Moser

1998

A natural generalization of the H enon map of the plane is a quadratic di eomorphism that has a quadratic inverse. We study the case when these maps are volume preserving, which generalizes the the family of symplectic quadratic maps studied by Moser. In this paper we obtain a characterization of these maps for dimension four and less. In addition, we use Moser's result to construct a subfamily of in n dimensions.

Quadratic volume-preserving maps

Nonlinearity, 1998

We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R 3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical "vortex-bubble". We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.