[Paper Review] Lipschitz invariance of walk dimension on connected self-similar sets

Hui Rao, Qingsong Gu|arXiv (Cornell University)|Sep 14, 2016

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper establishes that the walk dimension—defined via the critical exponent of Besov spaces—is invariant under bi-Lipschitz transformations on connected, self-similar sets equipped with an Ahlfors regular measure. By proving that the critical exponent β* is preserved under bi-Lipschitz maps, the authors show that distinct generalized Sierpiński gaskets with different β* values cannot be Lipschitz equivalent, providing a new invariant for distinguishing such fractals.

ABSTRACT

Walk dimension is an important conception in analysis of fractals. In this paper we prove that the walk dimension of a connected compact set possessing an Alfors regular measure is an invariant under Lipschitz transforms. As an application, we show some generalized Sierpiński gaskets are not Lipschitz equivalent.

Motivation & Objective

• To establish a new Lipschitz invariant for connected self-similar fractals using the critical exponent β* of Besov spaces.
• To prove that β* remains unchanged under bi-Lipschitz transformations on Ahlfors regular metric measure spaces.
• To apply this invariant to show non-Lipschitz equivalence of generalized Sierpiński gaskets with different walk dimensions.
• To connect the critical exponent β* to the walk dimension via heat kernel estimates and Dirichlet forms.
• To provide a method for distinguishing connected self-similar sets that are not bi-Lipschitz equivalent.

Proposed method

• Define the functional Wσ,M(u) measuring local L2 oscillation to construct the Besov-type space Wσ,2(M,d,μ).
• Introduce the critical exponent β* as the supremum of σ for which Wσ,2 contains non-constant functions.
• Use bi-Lipschitz invariance of volume doubling and Poincaré-type inequalities to show β* is preserved under bi-Lipschitz maps.
• Establish the equivalence between β* and the walk dimension β via heat kernel estimates satisfying the conditions in Grigor’yan, Hu, and Lau (2015).
• Compute β* for generalized Sierpiński gaskets using Dirichlet forms and (Δ-Y)-transforms to derive resistance scaling constants.
• Verify that the heat kernels satisfy the two-sided estimates required by Theorem 3.1, confirming β = β*.

Experimental results

Research questions

• RQ1 Is the critical exponent β* of Besov spaces invariant under bi-Lipschitz transformations on Ahlfors regular metric measure spaces?
• RQ2 Can β* serve as a Lipschitz invariant to distinguish non-bi-Lipschitz-equivalent self-similar sets?
• RQ3 Does the walk dimension β coincide with β* under mild conditions on the heat kernel?
• RQ4 Are generalized Sierpiński gaskets with different β* values necessarily non-Lipschitz equivalent?
• RQ5 Can the walk dimension be computed via Dirichlet forms and heat kernel estimates on self-similar fractals?

Key findings

• The critical exponent β* is invariant under bi-Lipschitz maps on connected, Ahlfors regular self-similar sets.
• The walk dimension β equals β* when the heat kernel satisfies the decay and integrability conditions in Theorem 3.1.
• For the Sierpiński gasket K1, β* = α + log(5/3)/log(2) ≈ α + 0.737.
• For the generalized gasket K2, β* = α + log(295/63)/log(8) ≈ α + 0.868.
• For K3 and K4, formed by composition of IFS, β* = α + log(1475/189)/log(16) ≈ α + 0.868, matching K2 but differing from K1.
• K1 is not Lipschitz equivalent to K2, K3, or K4 due to distinct β* values, while K3 and K4 are not distinguished by β* and may be Lipschitz equivalent.