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

Abstract

We examine the question whether the dimension $D$ of a set or probability measure is the same as the dimension of its image under a typical smooth function, if the range space is at least $D$-dimensional. If $\mu$ is a Borel probability measure of bounded support in ${\Bbb R}^n$ with correlation dimension $D$, and if $m\geq D$, then under almost every continuously differentiable function (‘almost every’ in the sense of prevalence) from ${\Bbb R}^n$ to ${\Bbb R}^m$, the correlation dimension of the image of $\mu$ is also $D$. If $\mu$ is the invariant measure of a dynamical system, the same is true for almost every delay coordinate map. That is, if $m\geq D$, then $m$ time delays are sufficient to find the correlation dimension using a typical measurement function. Further, it is shown that finite impulse response (FIR) filters do not change the correlation dimension. Analogous theorems hold for Hausdorff, pointwise, and information dimensions. We show by example that the conclusion fails for box-counting dimension.