Abstract
Given a categorical dynamical system, \textit{i.e.} a triangulated category together with an endofunctor, one can try to understand the complexity of the system by computing the entropy of the endofunctor. Computing the entropy of the composition of two endofunctors is hard, and in general the result doesn't have to be related to the entropy of the single pieces.In this paper we compute the entropy of the composition of two spherical twists around spherical objects, showing that it depends on the dimension of the graded vector space of morphisms between them. As a consequence of these computations we produce new counterexamples to Kikuta-Takahashi's conjecture. In particular, we describe the first counterexamples in odd dimension and examples for the $d$-Calabi-Yau Ginzburg dg algebra associated to the $A_2$ quiver.
Acknowledgments
The authors would like to thank Kohei Kikuta and Ed Segal for reading a draft of this preprint and providing many helpful suggestions. F.B. would like to thank his advisor Ed Segal for many helpful conversations, and Wendelin Lutz and Qaasim Shafi for a short conversation regarding Example 5.0.2. F.B. was supported by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No.725010). J.K. was supported by the Institute for Basic Science (IBS-R003-D1).
Citation
Federico Barbacovi. Jongmyeong Kim. "Entropy of the composition of two spherical twists." Osaka J. Math. 60 (3) 653 - 670, July 2023.
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