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MASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY

Published online by Cambridge University Press:  04 June 2020

AKISHI IKEDA*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka560-0043, Japan email ikeda@math.sci.osaka-u.ac.jp

Abstract

In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.

Type
Article
Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

This work is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, JSPS KAKENHI Grant Number JP16K17588, 16H06337, and JSPS bilateral Japan-Russia Research Cooperative Program. This paper was written while the author was visiting Perimeter Institute for Theoretical Physics by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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