Geometry and topology

On the set of subarcs in some non-postrcritically finite dendrites

N. V. Abrosimov^abc, M. V. Chanchieva^d, A. V. Tetenov<sup>ad

a</sup> Regional Scientific and Educational Mathematical Center, Tomsk State University, pr. Lenina, 36, 634050, Tomsk, Russia
^b Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
^c Novosibirsk State University, Pirogova str., 1, 630090, Novosibirsk, Russia
^d Gorno-Altaysk State University, Lenkina str., 1, 649000, Gorno-Altaysk, Russia

Abstract: We construct a family ${\mathbf F}$ of non-PCF dendrites $K$ in a plane, such that for any dendrite $K\in {\mathbf F}$ all its subarcs have the same Hausdorff dimension $s$, while the set of $s$-dimensional Hausdorff measures of subarcs connecting the given point and a self-similar Cantor subset in $K$ is a Cantor discontinuum.

Keywords: self-similar dendrite, ramification point, Hausdorff dimension, postcritically finite set.

UDC: 515.124, 519.172

MSC: 28A80

Received November 29, 2018, published August 3, 2019

Language: English

DOI: 10.17377/semi.2019.16.066

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