On metric properties of self-affine polygonal dendrites

TL;DR

This paper establishes a Hölder continuity property for Jordan arcs in self-affine polygonal dendrites, showing a specific diameter-to-distance relationship with constants C and λ.

Contribution

It proves a new metric property of self-affine dendrites generated by polygonal systems, specifically a Hölder condition for arcs within these fractals.

Findings

Jordan arcs satisfy a diameter bound with respect to endpoint distance.

Constants C and λ depend on the self-affine system.

The result applies to all points in the dendrite.

Abstract

We prove that for any self-affine dendrite K generated by a polygonal system, there are constants C>0 and $\lambda\in(0, 1)$ such that for any x, y in K, the Jordan arc $\gamma$ in K with endpoints x, y satisfies the inequality $diam({\gamma})\le C |x-y|^\lambda$.