TY - JOUR
T1 - Ergodic fractal measures and dimension conservation
AU - Furstenberg, Hillel
PY - 2008/4
Y1 - 2008/4
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=43449123351&partnerID=8YFLogxK
U2 - 10.1017/S0143385708000084
DO - 10.1017/S0143385708000084
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AN - SCOPUS:43449123351
SN - 0143-3857
VL - 28
SP - 405
EP - 422
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
IS - 2
ER -