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MASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY

Part of: Abelian categories Topological dynamics

Published online by Cambridge University Press:  04 June 2020

AKISHI IKEDA Open the ORCID record for AKISHI IKEDA [Opens in a new window]
AKISHI IKEDA*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka560-0043, Japan email ikeda@math.sci.osaka-u.ac.jp
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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.

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 37B40: Topological entropy
Type
Article
Information
Nagoya Mathematical Journal , Volume 244 , December 2021 , pp. 136 - 157
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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.

References

Adler, R., Konheim, A. and McAndrew, M., Topological entropy , Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential , Ann. Inst. Fourier (Grenoble) 59(6) (2009), 2525–2590.CrossRefGoogle Scholar
Arcara, D. and Bertram, A., Bridgeland-stable moduli spaces for K-trivial surfaces , J. Eur. Math. Soc. (JEMS) 15(1) (2013), 1–38; With an appendix by Max Lieblich.CrossRefGoogle Scholar
Assem, I., Simson, D. and Skowronski, A., “ Elements of the representation theory of associative algebras ”, in Techniques of Representation Theory, Vol. 1, London Mathematical Society Student Texts 65 , Cambridge University Press, Cambridge, 2006.Google Scholar
Bayer, A., A short proof of the deformation property of Bridgeland stability conditions, preprint, 2016, arXiv:1606.02169.Google Scholar
Beĭlinson, A. A., Bernstein, J. and Deligne, P., “ Faisceaux pervers ”, in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 100 , Soc. Math. France, Paris, 1982, 5–171.Google Scholar
Bondal, A., Representations of associative algebras and coherent sheaves , Izv. Akad. Nauk SSSR Ser. Mat. 53(1) (1989), 25–41; translation in Math. USSR-Izv., 34(1) (1990), 23–42.Google Scholar
Bridgeland, T., Stability conditions on triangulated categories , Ann. of Math. (2) 166(2) (2007), 317–345.CrossRefGoogle Scholar
Bridgeland, T., Stability conditions on K3 surfaces , Duke Math. J. 141(2) (2008), 241–291.CrossRefGoogle Scholar
Bridgeland, T. and Smith, I., Quadratic differentials as stability conditions , Publ. Math. Inst. Hautes Études Sci. 121 (2015), 155–278.CrossRefGoogle Scholar
Dimitrov, G., Haiden, F., Katzarkov, L. and Kontsevich, M., Dynamical systems and categories , Contemp. Math. 621 (2014), 133–170.CrossRefGoogle Scholar
Gaiotto, D., Moore, G. and Neitzke., A., Wall-crossing, Hitchin Systems, and the WKB Approximation , Adv. Math. 234 (2013), 239–403.Google Scholar
Ginzburg, V. G., Calabi-Yau algebras, preprint, 2006, arXiv:math/0612139.Google Scholar
Haiden, F., Katzarkov, L. and Kontsevich, M., Flat surfaces and stability structures , Publ. Math. Inst. Hautes Études Sci. 126 (2017), 247–318.CrossRefGoogle Scholar
Ikeda, A., Stability conditions on CYN categories associated to A n-quivers and period maps , Math. Ann. 367 (2017), 1–49.CrossRefGoogle Scholar
Keller, B., Deformed Calabi–Yau completions , J. Reine Angew. Math. 654 (2011), 125–180; With an appendix by Michel Van den Bergh.Google Scholar
Kikuta, K., On entropy for autoequivalences of the derived category of curves , Adv. Math. 308 (2017), 699–712.CrossRefGoogle Scholar
Kikuta, K. and Takahashi, A., On the categorical entropy and topological entropy , Int. Math. Res. Not. IMRN 2019(2) (2019), 457–469.CrossRefGoogle Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, preprint, 2008, arXiv:0811.2435.Google Scholar
Seidel, P. and Thomas, R. P., Braid group actions on derived categories of coherent sheaves , Duke Math. J. 108(1) (2001), 37–108.CrossRefGoogle Scholar
Shub, M., Dynamical systems, filtrations and entropy , Bull. Am. Math. Soc. 80 (1974), 27–41.Google Scholar
Yomdin, Y., Volume growth and entropy , Israel J. Math. 57 (1987), 285–300.CrossRefGoogle Scholar