UNIFORMLY BRANCHING TREES

Bonk, Mario and Meyer, Daniel (2022) UNIFORMLY BRANCHING TREES. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 375 (6). pp. 3841-3897.

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Official URL: http://dx.doi.org/10.1090/tran/8404

Abstract

<p>A quasiconformal tree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a (compact) metric tree that is doubling and of bounded turning. We call <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> trivalent if every branch point of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is uniformly branching. We prove that a metric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is quasisymmetrically equivalent to the continuum self-similar tree if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent.</p>

Item Type:	Article
Uncontrolled Keywords:	Mental Health
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	14 Sep 2022 14:49
Last Modified:	17 Mar 2024 14:30
DOI:	10.1090/tran/8404
Open Access URL:	https://arxiv.org/abs/2004.07912
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3164703