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Strong submeasures and applications to non-compact dynamical systems

Part of: Birational geometry

Published online by Cambridge University Press:  14 December 2020

TUYEN TRUNG TRUONG Open the ORCID record for TUYEN TRUNG TRUONG [Opens in a new window]
TUYEN TRUNG TRUONG*
Affiliation:
Department of Mathematics, University of Oslo, Blindern, 0851Oslo, Norway
*
e-mail:[email protected]
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Abstract

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A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

Keywords

invariant submeasuresopen-dense defined mapspullback and pushforwardstrong submeasurevariational principle

MSC classification

Secondary: 14E05: Rational and birational maps
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 287 - 309
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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