On the categorical entropy and the topological entropy

Entropy of an autoequivalence on Calabi-Yau manifolds

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.

Categorical vs topological entropy of autoequivalences of surfaces

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.

稳定流形中的轨道与范畴熵：有限 Albanese 态射簇的应用

Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms

本文证明具有有限 Albanese 态射的光滑射影簇满足 Gromov--Yomdin 性质，通过轨道逃逸原理和有限平展覆盖归约，并构造正范畴熵自等价，其存在性等价于非一般型。

关于 $\mathbb{P}$-twists 的熵

On entropy of $\mathbb{P}$-twists

本文通过计算范畴熵函数，证明光滑射影簇上 $\mathbb{P}$-对象诱导的 $\mathbb{P}$-twist 不共轭于标准自等价，并在附加条件下确定其与 spherical twist 的范畴多项式熵。

On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

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.

A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines

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.