General fractal dimensions of typical sets and measures

Abstract

Consider $(Y,\rho )$ as a complete metric space and $S$ as the space of probability Borel measures on Y. Let ${\bar{\dim }}_B^{\Psi ,\Phi }\left(E\right)$ be the general upper box dimension of the set $E\subset Y$. We begin by proving that the general packing dimension of the typical compact set, in the sense of the Baire category, is at least $\inf \left\{{\bar{\dim }}_B^{\Psi ,\Phi }\right(B(x,r)\left)\right|x\in Y,r>0\}$ where $B(x,r)$ is the closed ball in Y with center at x and radii $r>0$. Next, we obtain some estimates of the general upper and lower box dimensions of typical measures in the sense of the Baire category. Finally, we demonstrate that if $S$ is equipped with the weak topology and under some assumptions then the set of measures possessing the general upper and lower correlation dimension zero are residual. Furthermore, the general upper correlation dimension of typical measures (in the sense of the Baire category) is approximated through the general local lower and upper entropy dimensions of Y.