On metric properties of self-affine polygonal dendrites

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 $diam(\gamma )\le C\mathrm{\mid }x-y{\mathrm{\mid }}^{\lambda }$.