WIAS Preprint No. 58, (1993)

The Hausdorff dimension of certain attractors.


  • Bothe, Hans Guenter

2010 Mathematics Subject Classification

  • 58F12 28A78


  • Hyperbolic attractors, Hausdorff dimension, pressure




For the solid torus V = S1 x D2 and a C1 embedding ƒ : V → V given by ƒ(t,x1,x2) = (φ(t),λ1(t)·x1+z1(t), λ2(t)·x2+z2(t)) with dφ⁄dt > 1, 0 < λi(t) < 1 the attractor Λ= ∩i=0 ƒi (V) is a solenoid, and for each disk D(t) = {t} x D2 (t∈S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by dimH Λ (t) = max(p1,p2) (0.1) where the real numbers pi are characterized by the condition that the pressure of the function log λipi: S1 → ℝ with respect to the expanding mapping φ : S1 → S1 becomes zero. (There are two further characterizations of these numbers.) It is proved that (0.1) holds provided λ1, λ2 are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the space of all embeddings ƒ with sup λi < Θ-2 (Θ the mapping degree of φ) the subset of those ƒ which have an intrinsically transverse attractor Λ is open and dense with respect to the C1 topology.

Appeared in

  • Ergodic Theory Dynam. Systems 15 (1995), no. 3, pp. 449--474.

Download Documents