Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem Hata from 1985 asserts that every connected attractor an IFS is locally and path-connected. give quantitative strengthening Hata’s theorem. First we prove (1/ s )-Hölder path-connected, where similarity dimension IFS. Then show parameterized α)-Hölder curve for all α > . At endpoint, = , Remes 1998 already established self-similar sets Euclidean space satisfy open set condition are curves. In secondary result, how to promote Remes’ spaces, but this setting require have positive -dimensional Hausdorff measure lieu condition. To close paper, determine sharp exponents parameterizations class self-affine Bedford-McMullen carpets build sponges. An interesting phenomenon emerges setting. While optimal parameter ℝ n always at most ambient may be strictly greater than