Ergodic fractal measures and dimension conservation

Hillel Furstenberg

Research output: Contribution to journalArticlepeer-review

77 Scopus citations


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.

Original languageEnglish
Pages (from-to)405-422
Number of pages18
JournalErgodic Theory and Dynamical Systems
Issue number2
StatePublished - Apr 2008
Externally publishedYes


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