How projections affect the dimension spectrum of fractal measures

doi:10.1088/0951-7715/10/5/002

How projections affect the dimension spectrum of fractal measures

Brian R Hunt et al 1997 Nonlinearity 10 1031-1046 doi: 10.1088/0951-7715/10/5/002

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Brian R Hunt

and Vadim Yu Kaloshin

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742-2431, USA

Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA

Recommended by P Grassberger

Abstract. We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.