Brian R Hunt and Vadim Yu Kaloshin 1997 Nonlinearity 10 1031 doi:10.1088/0951-7715/10/5/002
Brian R Hunt
and Vadim Yu Kaloshin![]()
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.
02.30.Cj Measure and integration
02.50.-r Probability theory, stochastic processes, and statistics
60B05 Probability measures on topological spaces
Issue 5 (September 1997)
Received 8 January 1996, in final form 19 June 1997
Brian R Hunt and Vadim Yu Kaloshin 1997 Nonlinearity 10 1031
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