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.

Topological dimension of singular-hyperbolic attractors

2003

An {\em attractor} is a transitive set of a flow to which all positive orbit close to it converges. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \cite{MPP}. The geometric Lorenz attractor \cite{GW} is an example of a singular-hyperbolic attractor with topological dimension geq2\geq 2geq2. We shall prove that {\em all} singular-hyperbolic attractors on compact 3-manifolds have topological dimension geq2\geq 2geq2. The proof uses the methods in \cite{MP}.

BOUNDS FOR DIMENSION OF THE ATTRACTOR OF A SCALED IFS

Asian-European Journal of Mathematics, 2013

A hyperbolic iterated function system (IFS) consists of a complete metric space X together with a finite set of contraction mappings on X. In this paper, the notion of scaled IFS is defined and its existence conditions are examined. The relation between the similarity dimension of the attractors of a given homogeneous IFS and a scaled IFS and its dependency on the scaling factor are studied. A lower and upper bounds for the Hausdorff dimension of the attractor of a scaled IFS is obtained.

Dimensions of strange nonchaotic attractors

Physics Letters A, 1989

Strange nonchaotic attractors in two-dimensional maps exhibit zero Lyapunov exponent along one direction and negative along the other. Evidence is presented indicating that the capacity dimension of these attractors is two while their information dimension is one.

On Hausdorff dimension of oscillatory motions in three body problems

2011

We show that for the Sitnikov example and for the restricted planar circular 3–body problem the set of oscillatory motions often has maximal Hausdorff dimension. Also, we construct Newhouse domains for both problems. 1 Introduction and statement of the results One of the most famous results by Poincare is on non-integrability of the three body problem. The key part of the proof is construction of the homoclinic picture. This picture was at the origin of “chaos theory ” over a century ago. Later the results by Birkhoff and Smale gave a deep insight to the dynamics associated with the homoclinic picture and constructed a wide range

On Hausdorff dimension of oscillatory motions in Sitnikov problem

2010

We show that for the Sitnikov example the set of oscillatory motions is often has maximal Hausdorff dimension. This is a preliminary version of the paper; in the final version we intend to provide a similar result for the restricted planar circular 3–body problem as well.