Generic properties of compact metric spaces

Abstract

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.

On the Hausdorff and packing measures of typical compact metric spaces

“…The statement in Part (3) of Corollary 2.6 has recently been obtained by Rouyer [13]. However, since Part (1) in Corollary 2.6 shows that S is a subset of T , we deduce that the statement in Part (2) is stronger than Rouyer's result in Part (3).…”

“…(1) (saying that dim H (X) = 0 for a typical compact metric space X) has already been obtained by Rouyer [13] (see Theorem A in Section 1), we believe that it is instructive to present a simple proof based on Theorem 2.4.…”

“…Theorem A. [13] A typical compact metric space X ∈ K GH satisfies dim H (X) = dim B (X) = 0 , dim B (X) = ∞ .…”

On average Hewitt–Stromberg measures of typical compact metric spaces

“…We also note that Theorem A shows that the lower Hewitt-Stromberg measure (and hence the Hausdorff measure and the Hausdorff dimension) of a typical compact metric space is as small as possible and that the upper Hewitt-Stromberg measure (and hence the packing measure and the packing dimension) of a typical compact metric space is as big as possible. Other studies of typical compact sets, see [5,15,20], show the same dichotomy. For example, [20] proves that a typical compact metric space has lower box dimension equal to 0 and upper box dimension equal to ∞, and Gruber [5] and Myjak and Rudnicki [15] prove that if X is a metric space, then the lower box dimension of a typical compact subset of X is as small as possible and that the upper box dimension of a typical compact subset of X is (in many cases) as big as possible.…”

“…Indeed, many different aspects of this problem have been studied by several authors during the past 20 years, including [1,4,6,12] and the references therein, and the question also appears implicitly in [5]. While (almost) all previous work, including, for example, [1,[4][5][6], study fractal measures of typical compact subsets of a given complete metric space, this paper adopts a new and different viewpoint introduced very recently by Rouyer [20] and investigated further in [12], namely, we investigate typical compact metric spaces belonging to the Gromov-Hausdorff space K GH of all compact metric spaces. For example, in [12] the authors prove the following result about the fractal measures of a typical compact metric spaces belonging to the Gromov-Hausdorff space K GH .…”