An Exceptional Set in the Ergodic Theory of Expanding Maps on Manifolds

• Autores: A.G. Abercrombie, Ramachandran D. Nair
• Localización: Monatshefte für mathematik, ISSN 0026-9255, Vol. 148, Nº 1, 2006, págs. 1-17
• Idioma: inglés
• DOI: 10.1007/s00605-005-0391-3
• Texto completo no disponible (Saber más ...)
• Resumen

  • Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff¿s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class t for some t in (0,1], let O(x) denote the forward orbit closure of x and for a positive real number d and let E(x0, d) denote the set of points x in M such that the distance from x0 to O is at least d. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that if t = 1 and if t < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.