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A NOTE ON ENTROPY OF AUTO-EQUIVALENCES: LOWER BOUND AND THE CASE OF ORBIFOLD PROJECTIVE LINES

Published online by Cambridge University Press:  26 June 2018

KOHEI KIKUTA
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan email k-kikuta@cr.math.sci.osaka-u.ac.jp
YUUKI SHIRAISHI
Affiliation:
International college, Osaka University, Toyonaka, Osaka, 560-0043, Japan email shiraishi@cbcmp.icou.osaka-u.ac.jp
ATSUSHI TAKAHASHI
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan email takahashi@math.sci.osaka-u.ac.jp

Abstract

Entropy of categorical dynamics is defined by Dimitrov–Haiden–Katzarkov–Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov–Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

Type
Article
Copyright
© 2018 Foundation Nagoya Mathematical Journal

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References

Dimitrov, G., Haiden, F., Katzarkov, L. and Kontsevich, M., Dynamical systems and categories, Contem. Math. 621 (2014), 133170.Google Scholar
Fan, Y. W., Entropy of an autoequivalences Calabi–Yau manifolds, preprint, 2017, arXiv:1704.06957.Google Scholar
Gromov, M., On the entropy of holomorphic maps, Enseign. Math. 49 (2003), 217235.Google Scholar
Gromov, M., Entropy, homology and semialgebraic geometry, Astérisque 145–146 (1987), 225240.Google Scholar
Geigle, W. and Lenzing, H., “A class of weighted projective curves arising in representation theory of finite-dimensional algebras”, in Singularities, Representation of Algebras, and Vector Bundles, Springer Lecture Notes 1273, Springer, Berlin, 1987, 265297.Google Scholar
Ikeda, A., Mass growth of objects and categorical entropy, preprint, 2016, arXiv:1612.00995.Google Scholar
Ishibashi, Y., Shiraishi, Y. and Takahashi, A., A uniqueness theorem for Frobenius manifolds and Gromov–Witten theory for orbifold projective lines, J. Reine Angew. Math. 702 (2015), 143171.Google Scholar
Ishibashi, Y., Shiraishi, Y. and Takahashi, A., Primitive forms for affine cusp polynomials, preprint, 2012, arXiv:1211.1128.Google Scholar
Iwaki, K. and Takahashi, A., Stokes matrices for the quantum cohomologies of a class of orbifold projective lines, J. Math. Phys. 54 (2013), 101701.Google Scholar
Keating, A., Homological mirror symmetry for hypersurface cusp singularities, Selecta Math. 24 (2018), 14111452.Google Scholar
Kikuta, K., On entropy for autoequivalences of the derived category of curves, Adv. Math. 308 (2017), 699712.Google Scholar
Kikuta, K. and Takahashi, A., On the categorical entropy and the topological entropy, Int. Math. Res. Notices (2017), https://doi.org/10.1093/imrn/rnx131.Google Scholar
Lenzing, H. and Meltzer, H., The automorphism group of the derived category for a weighted projective line, Comm. Algebra 28 (2000), 16851700.Google Scholar
Lunts, V., Lefschetz fixed point theorems for Fourie–Mukai functors and dg algebras, J. Algebra 356 (2012), 230256.Google Scholar
Miyachi, J. and Yekutieli, A., Derived Picard groups of finite dimensional hereditary algebras, Compositio Math. 129 (2001), 341368.Google Scholar
Ouchi, G., Automorphisms of positive entropy on some hyperKähler manifolds via derived automorphisms of K3 surfaces, preprint, 2016, arXiv:1608.05627.Google Scholar
Ouchi, G., On entropy of spherical twists, preprint, 2017, arXiv:1705.01001.Google Scholar
Rossi, P., Gromov–Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations, Math. Ann. 348 (2010), 265287.Google Scholar
Shklyarov, D., Hirzebruch–Riemann–Roch theorem for DG algebras, preprint, 2007, arXiv:0710.1937v3.Google Scholar
Shiraishi, Y. and Takahashi, A., On the Frobenius manifolds for cusp singularities, Adv. Math. 273 (2015), 485522.Google Scholar
Takahashi, A., Weighted projective lines associated to regular systems of weights dual type, Adv. Stud. Pure Math. 59 (2010), 371388.Google Scholar
Takahashi, A., Mirror symmetry between orbifold projective lines and cusp singularities, Adv. Stud. Pure Math. 66 (2013), 257282.Google Scholar
Truong, T. T., Relations between dynamical degrees, Weil’s Riemann hypothesis and the standard conjectures, preprint, 2016, arXiv:1611.01124.Google Scholar
Ueda, K., Homological mirror symmetry and simple elliptic singularities, preprint, arXiv:math/0604361.Google Scholar
Yomdin, Y., Volume growth and entropy, Israel J. Math. 57 (1987), 285300.Google Scholar
Yoshioka, K., Categorical entropy for Fourier–Mukai transforms on generic abelian surfaces, preprint, 2017, arXiv:1701.04009.Google Scholar