How projections affect the dimension spectrum of fractal measures Hunt, Brian R Kaloshin, Vadim ; https://orcid.org/0000-0002-6051-2628 Mathematical Physics General Physics and Astronomy Applied Mathematics Statistical and Nonlinear Physics 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. IOP Publishing 1997 info:eu-repo/semantics/article doc-type:Article Article http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/8527 Hunt BR, Kaloshin V. How projections affect the dimension spectrum of fractal measures. <i>Nonlinearity</i>. 1997;10(5):1031-1046. doi:<a href="https://doi.org/10.1088/0951-7715/10/5/002">10.1088/0951-7715/10/5/002</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1088/0951-7715/10/5/002 info:eu-repo/semantics/altIdentifier/issn/0951-7715 info:eu-repo/semantics/altIdentifier/issn/1361-6544 info:eu-repo/semantics/closedAccess