Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T01:28:17.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Representation of families of sets by measures, dimension spectra and Diophantine approximation

Published online by Cambridge University Press:  01 January 2000

K. J. FALCONER
K. J. FALCONER
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland; e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A family of sets {Fd}d is said to be ‘represented by the measure μ’ if, for each d, the set Fd comprises those points at which the local dimension of μ takes some specific value (depending on d). Finding the Hausdorff dimension of these sets may then be thought of as finding the dimension spectrum, or multifractal spectrum, of μ. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approximation, Kleinian groups and hyperbolic dynamical systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 128 , Issue 1 , January 2000 , pp. 111 - 121
Copyright
The Cambridge Philosophical Society 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)