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Ergodic fractal measures and dimension conservation

Published online by Cambridge University Press:  01 April 2008

HILLEL FURSTENBERG
HILLEL FURSTENBERG*
Affiliation:
Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel (email: [email protected])
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Abstract

A linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For ‘homogeneous’ fractals (to be defined), there is a phenomenon of ‘dimension conservation’. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This ‘almost everywhere’ result implies a non-probabilistic statement for homogeneous fractals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 405 - 422
Copyright
Copyright © Cambridge University Press 2008

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