Lipschitz functions and approximate resolutions

Abstract

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.

Bi-Lipschitz maps and the category of approximate resolutions

“…For more details on approximate resolutions and Lipschitz maps, the reader is referred to [5] and [8,9], respectively.…”

“…Following the approach of Alexandroff and Urysohn (see [1] and [11,[2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]), given a space X and a normal sequence U on X, we define a metric d U on X.…”

“…The authors introduced a new method to study Lipschitz maps, using approximate resolutions in their earlier paper [8]. Given any compact metrizable spaces with an approximate resolution, there is an induced metric that gives the same uniformity, and Lipschitz maps between compact spaces with the so obtained metrics are studied by using approximate resolutions.…”

Lipschitz matchbox manifolds

“…The proof that Disp(ℓ ν , ν ℓ ) < ∞ implies Lipschitz equivalence for the metrics defined by (20) with a ℓ = 3 −ℓ is an exercise in the definitions, using the expression (17) for the metric on the fibers. The converse direction, that Lipschitz equivalence implies bounded tower equivalence, follows from the works of Miyata and Watanabe [108,109].…”

“…The presentation P of an inverse limit S P can be used to construct a "natural" metric on the space, and which is well-adapted to Lipschitz maps between such spaces. This has been studied in detail in the works by Miyata and Watanabe [108,109,110,111,112]. In the case of weak solenoids, this construction of natural metrics adapted to the resolution takes on a simplified form.…”