Lower bounds for the Hausdorff dimension of attractors (original) (raw)

On Hausdorff dimension for attractors of iterated function systems

Journal of the Australian Mathematical Society, 1993

A conjecture on the Hausdorff dimension for Markov attractors of disjoint hyperbolic iterated function systems was given by Ellis and Branton. This paper proves the conjecture and generalizes the result to more general cases.

On the existence of attractors

Transactions of the American Mathematical Society, 2012

On every compact 3-manifold, we build a non-empty open set U of Diff 1 (M ) such that, for every r ≥ 1, every C r -generic diffeomorphism f ∈ U ∩ Diff r (M ) has no topological attractors. On higher dimensional manifolds, one may require that f has neither topological attractors nor topological repellers. Our examples have finitely many quasi attractors. For flows, we may require that these quasi attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics. * This work has been done during the stays of Li Ming and Yang Dawei at the IMB, Université de Bourgogne and we thank the IMB for its warm hospitality. M. Li is supported by a post doctoral grant of the Région Bourgogne, and D. Yang is supported by CSC of Chinese Education Ministry.

Box dimension of trajectories of some discrete dynamical systems

Chaos, Solitons & Fractals, 2007

We study the asymptotics, box dimension, and Minkowski content of trajectories of some discrete dynamical systems. We show that under very general conditions, trajectories corresponding to parameters where saddle-node bifurcation appears have box dimension equal to 1/2, while those corresponding to period-doubling bifurcation parameter have box dimension equal to 2/3. Furthermore, all these trajectories are Minkowski nondegenerate. The results are illustrated in the case of logistic map.

Estimates of dimension of attractors of reaction-diffusion equations in the non-differentiable case

Comptes Rendus de l'Académie des Sciences- …, 1997

We improve a general theorem of 0. A. Ladyzhenskaya on the dimension of compact invariant sets in Hilbert spaces, and use it to prove that the dimension of global compact attractors of some differential inclusions and reaction-diffusion equations are finite. Estimations de la dimension des attracteurs des kquations de &action-diffusion dans ,le cas non-d@Zrentiel R&urn& Nous obtenons une ame'lioration d'un thPor2me g&Lral de 0. A. Ladyzhenskaya sur la dimension des ensembles compacts et invariants dans les espaces de Hilbert, et noun l'utilisons pour montrer que les dimerlsions des attracteurs globaux de quelques inclusions diff&entielles et celles des e'quations de r&action-diffusion sont jnies. Version francaise abrkgke Dans cette Note, now hdions les dimensisons de Hausdorff (&) et fractale (dr) des ensembles compacts et invariants A dans les espaces de Hilbert. Nous obtenons les rksultats suivants, oti H reprksente un espace de Hilbert, V : H-+ 17 une application de H dans H, et A un ensemble compact et invariant de V (d = V(d)).

Counterexamples to regularity of Mañé projections in the theory of attractors

This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least C 1 -smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C 1 -smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mañé's theorem it projects bijectively and Hölder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mañé projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.

Piecewise contracting maps on the interval: Hausdorff dimension, entropy, and attractors

Canadian Mathematical Bulletin

We consider the attractor Lambda\Lambda Lambda of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of Lambda\Lambda Lambda via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.