Lyapunov exponents of hyperbolic attractors

 Let μ ₊ be the SBR measure on a hyperbolic attractor Ω of a C ² Axiom A diffeomorphism (M,f) and v the volume measure on M. As is known, μ ₊ -almost every is Lyapunov regular and the Lyapunov characteristic exponents of (f,Df) at x are constants $\lambda^{(i)}(\mu_+,f),1\leq i\leq s$. In this paper we prove that $v$-almost every $x$ in the basin of attraction $W^s(\Omega)$ is positively regular and the Lyapunov characteristic exponents of $(f,Df)$ at $x$ are the constants . Similar results are also obtained for nonuniformly completely hyperbolic attractors. Received: 20 September 2001     

Lyapunov functions in the attractors dimension theory

The effectiveness of constructing Lyapunov functions in the attractors dimension theory is theory of the dimension demonstrated. Formulae for the Lyapunov dimension of the Lorenz, Hénon and Chirikov attractors are derived and proved. A hypothesis regarding the formula for the dimension of the Rössler attractor is formulated.     

Lyapunov functions and attractors in arbitrary metric spaces

We prove two theorems concerning Lyapunov functions on metric spaces. The new element in these theorems is the lack of a hypothesis of compactness or local compactness. The first theorem applies to a discrete dynamical system on any metric space; the result is that if is an attractor for a continuous map of a metric space to itself, then there is a Lyapunov function for . The second theorem applies only to separable metric spaces; the theorem is that there is a complete Lyapunov function for any continuously-generated discrete dynamical system on a separable metric space. (A complete Lyapunov function is a real-valued function that is constant on orbits in the chain recurrent set, is strictly decreasing along all other orbits, and separates different components of the chain recurrent set.)

     

Homogeneous polynomials with isomorphic milnor algebras

We recall first Mather’s Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. We show that two homogeneous polynomials having isomorphic Milnor algebras are right-equivalent.     

A subdivision algorithm for the computation of unstable manifolds and global attractors

Summary. Each invariant set of a given dynamical system is part of the global attractor. Therefore the global attractor contains all the potentially interesting dynamics, and, in particular, it contains every (global) unstable manifold. For this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically. In this article we develop such an algorithm using a subdivision technique. We prove convergence of this method in a very general setting, and, moreover, we describe the qualitative convergence behavior in the presence of a hyperbolic structure. The algorithm can successfully be applied to dynamical systems of moderate dimension, and we illustrate this fact by several numerical examples. Received May 11, 1995 / Revised version received December 6, 1995     

Constant milnor number implies constant multiplicity for quasihomogeneous singularities

We show that for certain topologically trivial deformations of an isolated hypersurface singularity the multiplicity does not change. This applies to all -constant first order deformations and to all -constant deformations of a quasihomogeneous singularity.Partially supported by the Deutsche Forschungsgemeinschaft and the National Science Foundation     

Bony attractors

A new possible geometry of an attractor of a dynamical system, a bony attractor, is described. A bony attractor is the union of two parts. The first part is the graph of a continuous function defined on a subset of ∑ ^k , the set of bi-infinite sequences of integers m in the range 0 ≤ m < k. The second part is the union of uncountably many intervals contained in the closure of the graph. An open set of skew products over the Bernoulli shift (σω) _i = ω _i+1 with fiber [0,1] is constructed such that each system in this set has a bony attractor.     

On intermediate attractors

It is proved, for a four-moment model of a phonon gas system and the Dirac–Schwindler extension of the Maxwell system, that a Chapman correct restriction of a initial-boundary problem exists. The well-posedness condition is found in terms of algebraic relations for parameters of the problem and elements of the boundary matrix.     

Multidimensional nonhyperbolic attractors

We construct smooth transformations and diffeomorphisms exhibiting nonuniformly hyperbolic attractors with multidimensional sensitiveness on initial conditions: typical orbits in the basin of attraction have several expanding directions. These systems also illustrate a new robust mechanism of sensitive dynamics: despite the nonuniform character of the expansion, the attractor persists in a full neighbourhood of the initial map. Partially supported by a J. S. Guggenheim Foundation Fellowship.     

Measure attractors and random attractors for stochastic partial differential equations

In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large     

Multidimensional Rovella-like attractors

We present a multidimensional flow exhibiting a Rovella-like attractor: a transitive invariant set with a non-Lorenz-like singularity accumulated by regular orbits and a multidimensional non-uniformly expanding invariant direction. Moreover, this attractor has a physical measure with full support and persists along certain sub-manifolds of the space of vector fields. As in the 3-dimensional Rovella-like attractor, this example is not robust. As a sub-product of the construction we obtain a new class of multidimensional non-uniformly expanding endomorphisms without any uniformly expanding direction, which is interesting by itself. Our example is a suspension (with singularities) of this multidimensional endomorphism.     

Multidimensional bony attractors

In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called a bone, and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., ??big.?? It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.