We study the categorical entropy and counterexamples to Gromov-Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan-Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston's classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.
We study categorical primitive forms for Calabi--Yau $A_\infty$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category ${\sf Fuk}(M)$ of a symplectic manifold $M$ has semi-simple Hochschild cohomology, then its genus zero Gromov--Witten invariants may be recovered from the $A_\infty$-category ${\sf Fuk}(M)$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enu…
We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and…
For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov-Haiden-Katzarkov-Kontsevich. We justify this notion by showing that for an automorphism of a smooth projective variety, the categorical polynomial entropy of the pullback functor on the derived category coincides with the polynomial growth rate of the induced actio…
We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the essential image of the right adjoint functor of the spherical functor contains a split-generator. We also see our results generalize the computations of the categorical entropy of spherical twists and $\mathbb{P}$-twists by Ouchi and Fan. As an application, we apply our results to the Gromov--Yomdin type conjecture by Kikuta--Takahashi.
To an exact endofunctor of a triangulated category with a split-generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. In this paper, we propose a conjecture which naturally generalizes the theorem by Gromov-Yomdin, and show that the categorical entropy of a surjective endomorphism of a smooth projective variety is equal to its topological entropy. Moreover, we compute the entropy of autoequivalences of the derived category in the case of the ample canonical or anti-canonical sheaf.
In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.
We prove that the categorical entropy of the autoequivalence $T_{\mathcal{O}}\circ(-\otimes\mathcal{O}(-1))$ on a Calabi-Yau manifold is the unique positive real number $λ$ satisfying $$ \sum_{k\geq 1}\frac{χ(\mathcal{O}(k))}{e^{kλ}}=e^{(d-1)t}. $$ We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.
In topological dynamics, the Gromov--Yomdin theorem states that the topological entropy of a holomorphic automorphism $f$ of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map $f^*$. In order to establish a categorical analogue of the Gromov--Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov--Yomdin type theorem holds for them.
Entropy of categorical dynamics is defined by Dmitrov-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.
In this paper, we compute categorical entropy of spherical twists. In particular, we prove that Gromov-Yomdin type conjecture holds for spherical twists. Moreover, we construct counterexamples of Gromov-Yomdin type conjecture for K3 surfaces modifying Fan's construction for even higher dimensional Calabi-Yau manifolds. The appendix, by Arend Bayer, shows non-emptyness of complements of a number of spherical objects in the derived categories of K3 surfaces.
Given a categorical dynamical system, 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 …
