Space filling curves and geodesic laminations

Abstract

In this paper we study a class of connected fractals that admit a space filling curve. We prove that these curves are Hölder continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.

Space-filling curves and geodesic laminations. II: Symmetries

“…In Ref. [18], we show there is a transversal measure to the lamination so that the map is expanding with respect to this transversal measure.…”

“…In Ref. [18] it is shown that the map ξ is 1/s-Hölder continuous when the IFS is self-similar, where s is the Hausdorff dimension of the attractor R. Moreover if the IFS {φ 1 , · · · , φ k } satisfies the open set condition and is self-similar then the map ξ is a measure preserving map between the Lebesgue measure of I and the normalized s-dimensional Hausdorff measure of R. If the IFS is not self-similar but s = d then ξ is measure preserving between the Lebesgue measure of I and the normalized Lebesgue measure of R ⊂ R d [18].…”

“…[18] was proved, using direct methods, that the associated lamination has a reflection symmetry, which corresponds to the symmetry along the diagonal that passes through (0, 0) and (1, 1) of R. This symmetry and the identity corresponds to the full group of symmetries of the partition {R 1 , . .…”

“…[18] the author introduced a class of space-filling curves associated to some connected fractals. These fractals are the fixed point of linear iterated function systems (IFS) satisfying a special property, known as the common point property, its definition is given in Sect.…”

“…The importance of this concept is that it allows to generalize the constructions of Ref. [18] to a wider class of fractals, that do not satisfy the common point property. In Corollary 5.1, we show that the symmetry group of the sub-lamination is a subgroup of the full group of symmetries of the attractor of the sub-IFS.…”