Skip to main content Accessibility help

Ergodic fractal measures and dimension conservation



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.



Hide All
[1]Doob, J. L.. Stochastic Processes. Wiley, New York, 1953.
[2]Furstenberg, H.. Intersections of Cantor sets and transversality of semigroups. Problems in Analysis (Princeton Mathematical Series, 31). Ed. R. C. Gunning. Princeton University Press, Princeton, NJ, 1970, pp. 4159.
[3]Furstenberg, H. and Weiss, B.. Markov processes and Ramsey theory for trees, Special Issue on Ramsey Theory. Combin. Probab. Comput. 12 (2003), 548563.
[4]Maker, P. T.. The ergodic theorem for a sequence of functions. Duke Math. J. 6 (1940), 2730.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed