Are the dimensions of a set and its image equal under typicalsmooth functions?

Abstract

We examine the question whether the dimension $D$ of a setor probability measure is the same as the dimension of itsimage under a typical smooth function, if the range spaceis at least $D$-dimensional. If $\mu$ is a Borelprobability measure of bounded support in ${\Bbb R}^n$with correlation dimension $D$, and if $m\geq D$, thenunder almost every continuously differentiable function(‘almost every’ in the sense of prevalence) from ${\BbbR}^n$ to ${\Bbb R}^m$, the correlation dimension of theimage of $\mu$ is also $D$. If $\mu$ is the invariantmeasure of a dynamical system, the same is true for almostevery delay coordinate map. That is, if $m\geq D$, then$m$ time delays are sufficient to find the correlationdimension using a typical measurement function. Further,it is shown that finite impulse response (FIR) filters donot change the correlation dimension. Analogous theoremshold for Hausdorff, pointwise, and information dimensions.We show by example that the conclusion fails forbox-counting dimension.