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How projections affect the dimension spectrum of fractal measures

Brian R Hunt-+ and Vadim Yu Kaloshin++

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Recommended by P Grassberger

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.


PACS

05.45.Df Fractals

02.30.Cj Measure and integration

02.50.-r Probability theory, stochastic processes, and statistics

MSC

60B05 Probability measures on topological spaces

28A80 Fractals (See also 37Fxx)

37D45 Strange attractors, chaotic dynamics

Subjects

Mathematical physics

Computational physics

Statistical physics and nonlinear systems

Dates

Issue 5 (September 1997)

Received 8 January 1996, in final form 19 June 1997



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