Hausdorff Dimension Of Invariant Sets Research Articles

Positive operators and Hausdorff dimension of invariant sets

In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are “infinitesimal similitudes” on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of $\sigma > 0$ for which $r(L_\sigma )=1$, where $L_\sigma$, $\sigma >0$, is a naturally associated family of positive linear operators and $r(L_\sigma )$ denotes the spectral radius of $L_\sigma$. We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

MULTIPLIERS OF HOMOCLINIC TANGENCIES AND A THEOREM OF GONCHENKO AND SHILNIKOV ON AREA PRESERVING MAPS

We investigate differential invariants for homoclinic tangencies and discuss the role of these invariants in the Hausdorff dimension of invariant sets associated with the tangency and its unfoldings. Further, we give a streamlined proof of a theorem of Goncheko and Shilnikov on the case of a tangency in an area preserving map. Specifically, the invariants determine whether or not a hyperbolic invariant set is formed near the tangency.

Upper and Lower Hausdorff Dimension Estimates for Invariant Sets ofk — 1 — Endomorphisms

For a special class of non–injective maps on Riemannian manifolds upper and lower bounds for the Hausdorff dimension of invariant sets are given in terms of the singular values of the tangent map. The upper estimation is based on a theorem by Douady and Oesterlé and its generalization to Riemannian manifolds by Noack and Reitmann, but additionally information about the noninjectivity is used. The lower estimation can be reached by modifying a method, derived by Shereshevskij for geometric constructions on the real line (also described by Barreira, for similar constructions in general metric spaces. The upper and lower dimension estimates for k — 1 — endomorphisms can for instance be applied to Julia sets of quadratic maps on the complex plane.

Hausdorff Dimension of Non-Hyperbolic Repellers. I: Maps with Holes

This is the first paper in a two-part series devoted to studying the Hausdorff dimension of invariant sets of non-uniformly hyperbolic, non-conformal maps. Here we consider a general abstract model, that we call piecewise smooth maps with holes. We show that the Hausdorff dimension of the repeller is strictly less than the dimension of the ambient manifold. Our approach also provides information on escape rates and dynamical dimension of the repeller.

Hausdorff dimension of repellors in low sensitive systems

Methods to estimate the Hausdorff dimension of invariant sets of scattering systems are presented. Based on the levels' hierarchical structure of the time delay function, these techniques can be used in systems whose future-invariant-set codimensions are approximately equal to or greater than one. The discussion is illustrated by a numerical example of a scatterer built with four hard spheres located at the vertices of a regular tetrahedron.

Finite-Dimensioned Models of Hydrodynamics Systems

We propose a general theory of non-linear systems of the fluid mechanical type, we present some of its applications in the mechanics of fluids, we pay our attention to the stochasticity of systems and we estimate the Hausdorff dimension of invariant sets of the Navier-Stokes equation.

On Hausdorff dimension of invariant sets for expanding maps of a circle

AbstractGiven an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).