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Conformal Geometry and Dynamics

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Metrics $ \rho$, quasimetrics $ \rho^s$ and pseudometrics $ \inf\rho^s$


Author: K. V. Storozhuk
Journal: Conform. Geom. Dyn. 21 (2017), 264-272
MSC (2010): Primary 54E35; Secondary 28A78, 30C62
DOI: https://doi.org/10.1090/ecgd/311
Published electronically: July 7, 2017
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Abstract: Let $ \rho $ be a metric on a space $ X$ and let $ s\geq 1$. The function $ \rho ^s(a,b)=\rho (a,b)^s$ is a quasimetric (it need not satisfy the triangle inequality). The function $ \inf \rho ^s(a,b)$ defined by the condition $ \inf \rho ^s(a,b)=\inf \{\sum _0^n \rho ^s(z_i,z_{i+1})$ $ z_0=a, z_n=b\}$ is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space $ (X,\rho )$. We also give some examples showing how the topology of the space  $ (X,\inf \rho ^s)$ can change as $ s$ changes.


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Additional Information

K. V. Storozhuk
Affiliation: Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia – and – Novosibirsk State University, 2, Pirogova Street, Novosibirsk 630090, Russia – and – RUDN University, 6 Miklukho-Makiaya st, Moscow, Russia, 117198
Email: stork@math.nsc.ru

DOI: https://doi.org/10.1090/ecgd/311
Received by editor(s): October 12, 2016
Received by editor(s) in revised form: April 17, 2017, May 7, 2017, and May 8, 2017
Published electronically: July 7, 2017
Additional Notes: This publication was supported by the Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008).
Article copyright: © Copyright 2017 American Mathematical Society

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