Abstract. Hurewicz's dimension-raising theorem states that for every n-to-1 map f : X → Y , dim Y ≤ dim X + n holds. In this paper we introduce a new notion of finite-to-one like map in a large scale setting. Using this notion we formulate a dimension-raising type theorem for the asymptotic dimension and the asymptotic Assouad-Nagata dimension. It is also well-known as Hurewicz's finite-to-one mapping theorem that dim X ≤ n if and only if there exists an (n + 1)-to-1 map from a 0-dimensional space onto X. We formulate a finite-to-one mapping type theorem for the asymptotic dimension and the asymptotic Assouad-Nagata dimension.

In an earlier paper the authors introduced a new approach using normal sequences and approximate resolutions to study Lipschitz maps between compact metric spaces. In this paper we intoduce two kinds of box-counting dimension, which are defined for every compact metric space with a normal sequence and for every approximate resolution of any compact metric space, and investigate their properties. In a special case those notions coincide with the usual box-counting dimension for compact subsets of R n . Our box-counting dimensions are Lipschitz subinvariant, where Lipschitz maps are in the sense of the earlier paper. Moreover, we obtain fundamental theorems such as the subset theorem, the product theorem and the sum theorem. As an example, for each r with 0 ≤ r ≤ ∞, we present a systematic way to construct a compact metric space with an approximate resolution whose box-counting dimension equals r.

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A Lipschitz function between metric spaces is an important notion in fractal geometry as it is well-known to have a close connection to fractal dimension. On the other hand, the theory of approximate resolutions has been developed by Mardešić and Watanabe. In this theory maps f : X → Y between general spaces are represented by approximate maps f : X → Y between approximate systems for any approximate resolutions p : X → X and q : Y → Y , and the approximate maps f give useful information about the properties of the maps f. In this paper, we describe a new method of using the theory of approximate resolutions to study Lipschitz functions. More precisely, first of all, given a Hausdorff space X and a normal sequence U with a reasonable condition, a new metric d U which induces the given topology is defined, and Lipschitz functions with respect to the metrics induced by normal sequences are characterized by a property of the normal sequences. Secondly, using this metric, for each compact metric space X and for each approximate resolution p : X → X of X with a reasonable condition, a new metric d p which is topologically equivalent to the given metric is defined, and the properties of those metrics are investigated. Lipschitz functions between continua with the metrics induced by approximate resolutions are characterized by approximate resolutions. As an application, contraction maps are characterized, and a sufficient condition in terms of approximate resolutions for the existence of a unique fixed point is obtained.

Abstract. This paper concerns the theory of approximate resolutions and its application to fractal geometry. In this paper, we first characterize a surjective map f : X → Y between compact metric spaces in terms of a property on any approximate map f : X → Y where p : X → X and q : Y → Y are any choices of approximate resolutions of X and Y , respectively. Using this characterization, we construct a category whose objects are approximate sequences so that the box-counting dimension, which was defined for approximate resolutions by the authors, is invariant in this category. To define the morphisms of the category, we introduce an equivalence relation on approximate maps and define the morphisms as the equivalence classes.