Self-similar sets, simple augmented trees, and their Lipschitz\n equivalence

2016/12/01 by Jun Luo, Luo, Jun Jason · 1 citation
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1612.00209

Abstract

Given an iterated function system (IFS) of contractive similitudes, the\ntheory of Gromov hyperbolic graph on the IFS has been established recently. In\nthe paper, we introduce a notion of simple augmented tree which is a Gromov\nhyperbolic graph. By generalizing a combinatorial device of rearrangeable\nmatrix, we show that there exists a near-isometry between the simple augmented\ntree and the symbolic space of the IFS, so that their hyperbolic boundaries are\nLipschitz equivalent. We then apply this to consider the Lipschitz equivalence\nof self-similar sets with or without assuming the open set condition. Moreover,\nwe also provide a criterion for a self-similar set to be a Cantor-type set\nwhich completely answers an open question raised in citeLaLu13. Our study\nextends the previous works.\n

Citations

• Fractal Geometry: Mathematical Foundations and Applications.

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• When the conformal dimension of a self-affine sponge of Lalley-Gatzouras type is zero

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