How projections affect the dimension spectrum of fractal measures

Abstract

We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension of the image of µ is min(m, D q (µ)); in particular, the q-dimension of µ is preserved provided m D q (µ). We also present results on the preservation of information dimension D 1 and pointwise dimension. Finally, for 0 q < 1 and q > 2 we give examples for which D q is not preserved by any linear transformation into R m. All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.

Multifractal concentrations of inertial particles in smooth random flows

“…Note that because of the convexity of n → ξ n , a necessary and sufficient condition to have multifractal clusters of particles (that is a non-linear dependence of the ζ n ) is that d H < d L and, of course, a value of the Stokes number below the critical clustering value. Hunt & Kaloshin (1997) showed that the dimension spectrum D n of fractal measures defined in the previous subsection is preserved under typical projections when 1 < n 2 ("typical projections" means here almost all of them). From a direct application of this result to the steady-state phase-space density f , one expects that for order-unity values of the Stokes number, ζ n = d−d H +min(n d, ξ n ) for 0 < n 1.…”

Stochastic suspensions of heavy particles

“…Ballistic motion for St ≫ 1 corresponds to D 2 → 2d, therefore a critical Stokes number St † exists such that D 2 (St † ) = d. The particles' spatial distribution is obtained by projecting the (2 × d)-dimensional phase space onto the d-dimensional physical space. It is tempting to apply a rigorous result on the projection of random fractal sets [22,42] stating that for almost all projections, the correlation dimension of the projected set is related to that of the unprojected one via the relation…”

Hausdorff dimension, projections, and the Fourier transform

“…Sauer and Yorke in [SY], and Hunt and Kaloshin in [HK1] and [HK2] investigated transformation of dimensions under typical smooth mappings, also in infinite dimensional spaces. Ledrappier and Lindenstrauss proved in [LL] a result analogous to Theorem 1.1 for measures on Riemann surfaces that are invariant under geodesic flow.…”