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

TL;DR

This paper investigates the entropy of auto-equivalences in categorical dynamics, establishing a lower bound generally and proving equality for orbifold projective lines, thus advancing understanding of entropy-spectral radius relations.

Contribution

It provides a general lower bound for the entropy-spectral radius equality and confirms this equality specifically for orbifold projective lines.

Findings

Established a lower bound for entropy in categorical dynamics.

Proved the entropy equals spectral radius for orbifold projective lines.

Enhanced understanding of entropy behavior in algebraic geometry contexts.

Abstract

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.

Tables

A	$(1,a_2,a_3)$	$(2,2,a_3)$	(2,3,3)	(2,3,4)	(2,3,5)
${\overrightarrow{\Delta }}_A$	${\tilde{A}}_{a_1,a_2}$	${\tilde{D}}_{a_3}$	${\tilde{E}}_6$	${\tilde{E}}_7$	${\tilde{E}}_8$

Equations

$$h(F)\ge \log \rho (\mathcal{N}(F)).$$

$$h(F)=\log \rho (\mathcal{N}(F)).$$

$${\delta }_{\mathcal{T},t}(M,N):=\{\begin{array}{ll}0& \text{ if }N\cong 0\\ \inf \left\{\sum _{i=1}^p\exp (n_{i}t)|\text{lx@xy@svg}\text{raise}0.0pt\text{raise}14.2263pt\text{raise}-2.9pt0\text{raise}14.2263pt\text{raise}-2.263pt{A}_1\text{raise}14.2263pt\text{raise}-2.25pt\dots \text{raise}14.2263pt\text{raise}-1.773pt{A}_{p-1}\text{raise}14.2263pt\text{raise}-3.4ptN\oplus N^{\prime }\text{raise}-14.2263pt\text{raise}-2.25ptM[n_1]\text{raise}-14.2263pt\text{raise}-2.25pt\dots \text{raise}-14.2263pt\text{raise}-2.25ptM[n_p]\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@droprule}\text{ignorespaces}\text{raise}14.2263pt\text{raise}0.0pt\text{lx@xy@tip}1\text{lx@xy@tip}-1\text{ignorespaces}\text{lx@xy@droprule}\text{lx@xy@droprule}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{raise}-9.7263pt\text{raise}0.0pt\text{lx@xy@tip}1\text{lx@xy@tip}-1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{raise}11.9785pt\text{raise}0.0pt\text{lx@xy@tip}1\text{lx@xy@tip}-1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@droprule}\text{ignorespaces}\text{raise}14.2263pt\text{raise}0.0pt\text{lx@xy@tip}1\text{lx@xy@tip}-1\text{ignorespaces}\text{lx@xy@droprule}\text{lx@xy@droprule}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{raise}-9.7263pt\text{raise}0.0pt\text{lx@xy@tip}1\text{lx@xy@tip}-1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{raise}9.84932pt\text{raise}0.0pt\text{lx@xy@tip}1\text{lx@xy@tip}-1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\text{lx@xy@drawline@}\text{ignorespaces}\right\}& \text{ if }N\in ⟨M⟩\\ \infty & \text{ if }N\notin ⟨M⟩.\end{array}$$

$${\delta }_{{\mathcal{D}}^b(\mathbb{C}),t}(\mathbb{C},M)=\sum _{l\in \mathbb{Z}}\left({\dim }_{\mathbb{C}}{H}^l(M)\right)\cdot e^{-lt}.$$

$$h_t(F):=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathcal{T},t}(G,F^{n}G).$$

$$h_t(F)=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathcal{T},t}(G,F^n{G}^{\prime }).$$

$$h_t(F)=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathrm{per}(B),t}^{\prime }(G,F^n{G}^{\prime }),$$

$${\delta }_{\mathrm{per}(B),t}^{\prime }(M,N):=\sum _{m\in \mathbb{Z}}{\dim }_{\mathbb{C}}{\mathrm{Hom}}_{\mathrm{per}(B)}(M,N[m])e^{-mt},M,N\in \mathrm{per}(B).$$

$$C_1(t){\delta }_{\mathrm{per}(B),t}(G,M)\le {\delta }_{\mathrm{per}(B),t}^{\prime }(G,M)\le C_2(t){\delta }_{\mathrm{per}(B),t}(G,M).$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathrm{per}(B),t}(G,M)=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathrm{per}(B),t}^{\prime }(G,M).$$

$$h_t(F)=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathrm{per}(B),t}(G,F^n{G}^{\prime })=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\delta }_{\mathrm{per}(B),t}^{\prime }(G,F^n{G}^{\prime }).$$

$$\chi (M,N):=\sum _{n\in \mathbb{Z}}(-1{)}^n{\dim }_{\mathbb{C}}{\mathrm{Hom}}_{\mathrm{per}(B)}(M,N[n]).$$

$$h(F)\ge \log \rho (\mathcal{N}(F)).$$

$$\Vert v\Vert :=\sum _{i=1}^p|{\chi }_{\mathbb{R}}(v_i,v)|,v\in \mathcal{N}(\mathrm{per}(B){)}_{\mathbb{R}},$$

$$\sum _{i,j=1}^p|\chi (v_i,\mathcal{N}(F)v_j)|\ge C\cdot \Vert \mathcal{N}(F{)}_{\mathbb{R}}\Vert .$$

$$h(F)$$

$$h(F)=\log \rho (\mathcal{N}(F)).$$

$$h(F)$$

$$h(F)=\log \rho (\mathcal{N}(F))=0.$$

$$\mathrm{Auteq}(\mathrm{per}(B))\cong ⟨{\mathcal{S}}_B,{\mathcal{S}}_B[-1]⟩\times \mathrm{Aut}(\vec{\Delta }),$$

$$R_{A,\Lambda }:=\mathbb{C}[X_1,\dots ,X_r]/I_{\Lambda },$$

$$X_i^{a_i}-X_2^{a_2}+{\lambda }_i{X}_1^{a_1},i=3,\dots ,r.$$

$$L_A:=\underset{i=1}{\overset{r}{⨁}}\mathbb{Z}{\vec{X}}_i/M_A,$$

$$a_i{\vec{X}}_i-a_j{\vec{X}}_j,1\le i<j\le r.$$

$$a:=l.c.m(a_1,\dots ,a_r),{\mu }_A:=2+\sum _{i=1}^r(a_i-1),{\chi }_A:=2+\sum _{i=1}^r(\frac{1}{a_i}-1).$$

$${\mathbb{P}}_{A,\Lambda }^1:=\left[\left(\mathrm{Spec}(R_{A,\Lambda })\backslash \{0\}\right)/\mathrm{Spec}(\mathbb{C}{L}_A)\right],$$

$$\mathrm{Coh}({\mathbb{P}}_{A,\Lambda }^1)={\mathrm{gr}}^{L_A}(R_{A,\Lambda })/{\mathrm{tor}}^{L_A}(R_{A,\Lambda }).$$

$${\mathcal{O}}_{{\mathbb{P}}_{A,\Lambda }^1}(\vec{l}):=[R_{A,\Lambda }(\vec{l})]\in \mathrm{Coh}({\mathbb{P}}_{A,\Lambda }^1),$$

$$\vec{x}=l\vec{c}+\sum _{i=1}^r{p}_i{\vec{x}}_i,0\le p_i\le a_i-1.$$

$$\mathrm{Hom}({\mathcal{O}}_{{\mathbb{P}}_{A,\Lambda }^1}(\vec{x}),{\mathcal{O}}_{{\mathbb{P}}_{A,\Lambda }^1}(\vec{y}))=0.$$

$${\mathrm{Ext}}^1(M_2,M_1)\cong \mathrm{Hom}(M_1,M_2\otimes {\mathcal{O}}_{{\mathbb{P}}_{A,\Lambda }^1}(\vec{\omega }){)}^{\ast }.$$

Full text

In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

1. Introduction

It is interesting to bring some dynamical view points into the category theory. Motivated by the classical theory of dynamical systems, the notion of entropy of categorical dynamical systems (entropy of endo-functors for short) is defined by Dimitrov–Haiden–Katzarkov–Kontsevich [DHKK]. The entropy of endo-functors is actually similar to the topological entropy in the sense of sharing many properties (Lemma 2.7, 2.8, 2.9). Moreover, the entropy of the derived pull-back of a surjective endomorphism of a smooth projective variety over ${\mathbb{C}}$ is equal to its topological entropy [KT]. In other words, the entropy of endo-functors can be thought of as a categorical generalization of the topological entropy.

In this paper, we add two results on the entropy of endo-functors. The first one is that, for the perfect derived categories ${\rm per}(B)$ of a smooth proper differential graded algebra $B$, the lower bound of the entropy $h(F)$ of an endo-functor $F$ is given by the natural logarithm of the spectral radius $\rho({\mathcal{N}}(F))$ on the numerical Grothendieck group, called the (numerical) Gromov–Yomdin type inequality (See also [KT, Conjecture 5.3]). It is motivated by the fundamental theorem of the topological entropy for complex dynamics on algebraic varieties due to Gromov–Yomdin [Gro1, Gro2, Yom]:

Theorem** (Theorem 2.13).**

For each endo-functor $F$ of ${\rm per}(B)$ admitting left or right adjoint functors, such that $F^{n}B\not\cong 0$ for $n\geq 0$, we have

[TABLE]

For the proof, we use some norm inspired by the theory of dynamical degree and algebraic cycles due to Truong [Tru]. Ikeda shows this inequality by the mass-growth for Bridgeland’s stability conditions [Ike].

The equality in the Gromov–Yomdin type inequality is now known to hold for elliptic curves [Kik], varieties with the ample (anti-)canonical sheaf [KT] and abelian surfaces [Yos], which gives some applications to the topological entropy of dynamics on moduli spaces of stable objects in the sense of Bridgeland [Ouc1, Yos]. But, in general, it does not hold for some Calabi-Yau varieties [Fan, Ouc2]. As a corollary of the first main theorem, it is easy to show the equality for derived categories of hereditary finite dimensional algebras (Proposition 2.14, Corollary 2.15).

The second result of this paper claims the equality for the derived category ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$ of an orbifold projective line ${\mathbb{P}}^{1}_{A,\Lambda}$ introduced by Geigle–Lenzing [GL]. Orbifold projective lines are important and interesting objects since they are not only in the next class to hereditary finite dimensional algebras but few examples whose homological and classical mirror symmetry are well-understood (cf. [IST, IST2, IT, Kea, Ros, ST, Tak1, Tak2, Ued]):

Theorem** (Theorem 3.10).**

For each auto-equivalence $F$ of ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$, we have

[TABLE]

Moreover, $\rho({\mathcal{N}}(F))$ is an algebraic number and $h(F)=0$ if $\chi_{A}\neq 0$.

It is an important and interesting problem to find a characterization of endo-functors attaining the lower bound of the inequality (1.1).

Acknowledgements. The first named author is supported by JSPS KAKENHI Grant Number JP17J00227. The second named author is supported by Research Fellowship of Japan Society for the Promotion for Young Scientists. The third named author is supported by JSPS KAKENHI Grant Number JP16H06337, JP26610008.

2. Preliminaries

2.1. Notations and terminologies

Throughout this paper, we work over the base field ${\mathbb{C}}$ and all triangulated categories are ${\mathbb{C}}$-linear and not equivalent to the zero category. The translation functor on a triangulated category is denoted by $[1]$. All (triangulated) functors are ${\mathbb{C}}$-linear.

A triangulated category ${\mathcal{T}}$ is called split-closed if every idempotent in ${\mathcal{T}}$ splits, namely, if it contains all direct summands of its objects, and it is called thick if it is split-closed and closed under isomorphisms. For an object $M\in{\mathcal{T}}$, we denote $\langle M\rangle$ by the smallest thick triangulated subcategory containing $M$. An object $G\in{\mathcal{T}}$ is called a split-generator if $\langle G\rangle={\mathcal{T}}$. A triangulated category ${\mathcal{T}}$ is said to be of finite type if for all $M,N\in{\mathcal{T}}$ we have $\sum_{n\in{\mathbb{Z}}}\dim_{\mathbb{C}}{\rm Hom}_{{\mathcal{T}}}(M,N[n])<\infty$.

2.2. Complexity

From now on, ${\mathcal{T}},{\mathcal{T}}^{\prime}$ denote triangulated categories of finite type.

Definition 2.1** (Definition 2.1 in [DHKK]).**

For each $M,N\in{\mathcal{T}}$, define the function $\delta_{{\mathcal{T}},t}(M,N):{\mathbb{R}}\longrightarrow{\mathbb{R}}_{\geq 0}\cup\{\infty\}$ in $t$ by

[TABLE]

The function $\delta_{{\mathcal{T}},t}(M,N)$ is called the complexity of $N$ with respect to $M$.

Remark 2.2*.*

If ${\mathcal{T}}$ has a split-generator $G$ and $M\in{\mathcal{T}}$ is not isomorphic to a zero object, then an inequality $1\leq\delta_{{\mathcal{T}},0}(G,M)<\infty$ holds.

We recall some basic properties of the complexity.

Lemma 2.3**.**

Let $M_{1},M_{2},M_{3},M_{4}\in{\mathcal{T}}$.

1. $(\rm{i})$

If $M_{2}\in\langle M_{1}\rangle$ and $M_{2}\not\cong 0$, then $0<\delta_{{\mathcal{T}},t}(M_{1},M_{2})$. 2. $(\rm{ii})$

If $M_{1}\cong M_{3}$, then $\delta_{{\mathcal{T}},t}(M_{1},M_{2})=\delta_{{\mathcal{T}},t}(M_{3},M_{2})$. 3. $(\rm{iii})$

If $M_{2}\cong M_{3}$, then $\delta_{{\mathcal{T}},t}(M_{1},M_{2})=\delta_{{\mathcal{T}},t}(M_{1},M_{3})$. 4. $(\rm{iv})$

If $M_{2}\not\cong 0$, then $\delta_{{\mathcal{T}},t}(M_{1},M_{3})\leq\delta_{{\mathcal{T}},t}(M_{1},M_{2})\delta_{{\mathcal{T}},t}(M_{2},M_{3})$. 5. $(\rm{v})$

We have $\delta_{{\mathcal{T}},t}(M_{4},M_{2})\leq\delta_{{\mathcal{T}},t}(M_{4},M_{1})+\delta_{{\mathcal{T}},t}(M_{4},M_{3})$ for an exact triangle $M_{1}\to M_{2}\to M_{3}$. 6. $(\rm{vi})$

We have $\delta_{{\mathcal{T}}^{\prime},t}(F(M_{1}),F(M_{2}))\leq\delta_{{\mathcal{T}},t}(M_{1},M_{2})$ for any triangulated functor $F:{\mathcal{T}}\longrightarrow{\mathcal{T}}^{\prime}$.

Lemma 2.4**.**

Let ${\mathcal{D}}^{b}({\mathbb{C}})$ be the bounded derived category of finite dimensional ${\mathbb{C}}$-vector spaces. For $M\in{\mathcal{D}}^{b}({\mathbb{C}})$, we have the following inequality

[TABLE]

2.3. Entropy of endo-functors

Endo-functor $F$ means triangulated functor $F:{\mathcal{T}}\to{\mathcal{T}}$. We assume that all endo-functors of ${\mathcal{T}}$ satisfy that $F^{n}G\not\cong 0$ for $n\geq 0$ (if ${\mathcal{T}}$ has a split-generator $G$).

Definition 2.5** (Definition 2.4 in [DHKK]).**

Let $G$ be a split-generator of ${\mathcal{T}}$ and $F$ an endo-functor of ${\mathcal{T}}$. The entropy of $F$ is the function $h_{t}(F):{\mathbb{R}}\longrightarrow\{-\infty\}\cup{\mathbb{R}}$ given by

[TABLE]

It follows from [DHKK, Lemma 2.5] that the entropy is well-defined and doesn’t depend on the choice of split-generators.

Lemma 2.6**.**

Let $G,G^{\prime}$ be split-generators of ${\mathcal{T}}$ and $F$ an endo-functor of ${\mathcal{T}}$. The entropy $h_{t}(F)$ of $F$ is given by

[TABLE]

The three lemmas below show that the entropy of endo-functors is similar to the topological entropy.

Lemma 2.7**.**

Let $G$ be a split-generator of ${\mathcal{T}}$ and $F_{1},F_{2}$ endo-functors of ${\mathcal{T}}$.

1. $(\rm{i})$

If $F_{1}\cong F_{2}$, then $h_{t}(F_{1})=h_{t}(F_{2})$. 2. $(\rm{ii})$

We have $h_{t}(F_{1}^{m})=mh_{t}(F_{1})$ for $m\geq 1$. 3. $(\rm{iii})$

we have $h_{t}(F_{1}F_{2})=h_{t}(F_{2}F_{1})$. 4. $(\rm{iv})$

If $F_{1}F_{2}\cong F_{2}F_{1}$, then $h_{t}(F_{1}F_{2})\leq h_{t}(F_{1})+h_{t}(F_{2})$. 5. $(\rm{v})$

If $F_{1}=F_{2}[m]~{}(m\in{\mathbb{Z}})$, then $h_{t}(F_{1})=h_{t}(F_{2})+mt$.

Lemma 2.8**.**

Let $F_{i}$ be an endo-functor of ${\mathcal{T}}_{i}$ with a split-generator $G_{i}~{}(i=1,2)$. If there exists a fully faithful functor $F^{\prime}:{\mathcal{T}}_{2}\to{\mathcal{T}}_{1}$, which has left and right adjoint functors, such that $F^{\prime}F_{2}\simeq F_{1}F^{\prime}$, then $h_{t}(F_{2})\leq h_{t}(F_{1})$.

Lemma 2.9**.**

Let $F_{i}$ be an endo-functor of ${\mathcal{T}}_{i}$ with a split-generator $G_{i}~{}(i=1,2)$. If there exists a essentially surjective functor $F^{\prime}:{\mathcal{T}}_{1}\to{\mathcal{T}}_{2}$ such that $F^{\prime}F_{1}\simeq F_{2}F^{\prime}$, then $h_{t}(F_{2})\leq h_{t}(F_{1})$.

As a corollary of Lemma 2.9, we have the following

Corollary 2.10**.**

Let $F^{\prime}$ an auto-equivalence of ${\mathcal{T}}$. The entropy is a class function, namely, $h_{t}(F^{\prime}FF^{\prime-1})=h_{t}(F)$.

Let $B$ be a smooth proper differential graded (dg) ${\mathbb{C}}$-algebra $B$ and ${\rm per}(B)$ the perfect derived category of dg $B$-modules, the full triangulated subcategory of the derived category ${\mathcal{D}}(B)$ of dg $B$-modules containing $B$ closed under isomorphisms and taking direct summands. By definition, $B$ is a split-generator of ${\rm per}(B)$.

The following proposition enables us to compute entropy.

Proposition 2.11** (Theorem 2.7 in [DHKK]).**

Let $G,G^{\prime}$ be split-generators of ${\rm per}(B)$ and $F$ an endo-functor of ${\rm per}(B)$. The entropy $h_{t}(F)$ is given by

[TABLE]

where

[TABLE]

Proof*.*

The following is proven in the proof of [DHKK, Theorem 2.7].

Lemma 2.12**.**

For each $M\in{\rm per}(B)$, there exist $C_{1}(t),C_{2}(t)$ for $t\in{\mathbb{R}}$ such that

[TABLE]

In particular, for each $M\in{\rm per}(B)$ we have

[TABLE]

Together with Lemma 2.6, we have

[TABLE]

We finished the proof of the proposition. ∎

In order to state the first main theorem, we prepare some terminologies. For $M,N\in{\rm per}(B)$, set

[TABLE]

It naturally induces a bilinear form on the Grothendieck group $K_{0}({\rm per}(B))$ of ${\rm per}(B)$, called the Euler form, which is denoted by the same letter $\chi$. Then the numerical Grothendieck group ${\mathcal{N}}({\rm per}(B))$ is defined as the quotient of $K_{0}({\rm per}(B))$ by the radical of $\chi$ (which is well-defined by the Serre duality). It is important to note that ${\mathcal{N}}({\rm per}(B))$ is a free abelian group of finite rank by Hirzebruch-Riemann-Roch theorem [Shk, Lun]. If an endo-functor $F$ of ${\rm per}(B)$ admits left or right (hence both by the Serre duality) adjoint functors, it respects the radical of $\chi$. Therefore, it induces an endomorphism ${\mathcal{N}}(F)$ on ${\mathcal{N}}({\rm per}(B))$. Note that an endo-functor lifting to a dg endo-functor of the dg category ${\rm per}_{dg}(B)$ admits adjoint functors. The spectral radius $\rho({\mathcal{N}}(F))$ of ${\mathcal{N}}(F)$ is the maximum of absolute values of eigenvalues of ${\mathbb{C}}$-linear endomorphism ${\mathcal{N}}(F)\otimes_{\mathbb{Z}}{\mathbb{C}}$. Set $\delta_{\mathcal{T}}:=\delta_{{\mathcal{T}},0},\delta^{\prime}_{\mathcal{T}}:=\delta^{\prime}_{{\mathcal{T}},0},h:=h_{0}$.

Inspired by the theory of dynamical degree and algebraic cycles due to Truong (cf. [Tru, eq. (3.2)]), we show the following:

Theorem 2.13**.**

For each endo-functor $F$ of ${\rm per}(B)$ admitting left or right adjoint functors, we have

[TABLE]

Proof*.*

Let $v_{1},\cdots,v_{p}~{}(v_{i}=[M_{i}],M_{i}\in{\rm per}(B))$ be a fixed basis of ${\mathcal{N}}({\rm per}(B))$. Set $M_{0}:=\oplus_{i}M_{i},{\mathcal{N}}({\rm per}(B))_{{\mathbb{R}}}:={\mathcal{N}}({\rm per}(B))\otimes_{{\mathbb{Z}}}{\mathbb{R}},{\mathcal{N}}(F)_{{\mathbb{R}}}:={\mathcal{N}}(F)\otimes_{{\mathbb{Z}}}{\mathbb{R}},\chi_{{\mathbb{R}}}:=\chi\otimes_{\mathbb{Z}}{\mathbb{R}}$. Define a norm $\|\cdot\|$ on ${\mathcal{N}}({\rm per}(B))_{\mathbb{R}}$ by

[TABLE]

which induces an operator norm of ${\mathcal{N}}(F)_{{\mathbb{R}}}$, that is, $\|{\mathcal{N}}(F)_{{\mathbb{R}}}\|:=\displaystyle\sup_{\|v\|=1}\|{\mathcal{N}}(F)_{{\mathbb{R}}}v\|$. By the compactness of the subset $\{\|v\|=1\}\subset{\mathcal{N}}({\rm per}(B))_{{\mathbb{R}}}$, there exists a positive number $C>0$ such that

[TABLE]

Note that $B\oplus M_{0}$ is a split-generator of ${\rm per}(B)$. By Proposition 2.11, the statement follows from

[TABLE]

∎

Let ${\rm Auteq}({\mathcal{T}})$ be the group of (natural isomorphism classes of) auto-equivalences of a triangulated category ${\mathcal{T}}$.

Proposition 2.14**.**

Let $B$ be a hereditary finite dimensional ${\mathbb{C}}$-algebra. For each auto-equivalence $F\in{\rm Auteq}({\rm per}(B))$, we have

[TABLE]

Proof*.*

Due to Theorem 2.13, we only need to show the upper bound. Let $P_{1},\dots,P_{\dim_{{\mathbb{C}}}B}$ be indecomposable modules. Each auto-equivalence $F$ sends an indecomposable object to an indecomposable one. Since $B$ is hereditary, there exists $m\in{\mathbb{Z}}$ such that the indecomposable object $F^{n}(P_{i})[m]$ is isomorphic to an object concentrated in degree zero, namely, a $B$-module. By Proposition 2.11, we have

[TABLE]

∎

Corollary 2.15**.**

Suppose that $B={\mathbb{C}}\vec{\Delta}$ for some Dynkin quiver $\vec{\Delta}$. Then, we have

[TABLE]

Proof*.*

It is known by [MY, Theorem 3.8], that

[TABLE]

where ${\mathcal{S}}_{B}$ is the Serre functor of ${\rm per}(B)$ and ${\rm Aut}(\vec{\Delta})$ is the finite subgroup of ${\rm Auteq}({\rm per}(B))$ consisting of automorphisms of $\vec{\Delta}$. Again, by [MY, Theorem 3.8], ${\mathcal{S}}_{B}$ is of finite order up to translation. The statement follows from Lemma 2.7 (ii), (iv) and (v). ∎

3. Orbifold projective lines

In this section, we shall show the Gromov–Yomdin type theorem for the entropy of an auto-equivalence on the derived category ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$ of coherent sheaves on an orbifold projective line ${\mathbb{P}}^{1}_{A,\Lambda}$. We first recall the definition of orbifold projective line in [GL].

Let $r\geq 3$ be a positive integer. Let $A=(a_{1},\ldots,a_{r})$ be a multiplet of positive integers and $\Lambda=(\lambda_{1},\ldots,\lambda_{r})$ a multiplet of pairwise distinct elements of ${\mathbb{P}}^{1}({\mathbb{C}})$ normalized such that $\lambda_{1}=\infty$, $\lambda_{2}=0$ and $\lambda_{3}=1$.

In order to introduce an orbifold projective line, we prepare some notations.

Definition 3.1**.**

Let $r$, $A$ and $\Lambda$ be as above.

1. $(\rm{i})$

Define a ring $R_{A,\Lambda}$ by

[TABLE] 2. $(\rm{ii})$

Denote by $L_{A}$ an abelian group generated by $r$-letters $\vec{X_{i}}$, $i=1,\dots,r$ defined as the quotient

[TABLE] 3. $(\rm{iii})$

Set

[TABLE]

We then consider the following quotient stack$:$

Definition 3.2**.**

Let $r$, $A$ and $\Lambda$ be as above. Define a stack ${\mathbb{P}}^{1}_{A,\Lambda}$ by

[TABLE]

which is called the orbifold projective line of type $(A,\Lambda)$.

The orbifold projective line is a Deligne–Mumford stack whose coarse moduli space is a smooth projective line ${\mathbb{P}}^{1}$.

Denote by ${\rm gr}^{L_{A}}(R_{A,\Lambda})$ the abelian category of finitely generated $L_{A}$-graded $R_{A,\Lambda}$-modules and denote by ${\rm tor}^{L_{A}}(R_{A,\Lambda})$ the full subcategory of ${\rm gr}^{L_{A}}(R_{A,\Lambda})$ whose objects are finite-dimensional $L_{A}$-graded $R_{A,\Lambda}$-modules. It is known (cf. [GL, Section 1.8]) that the abelian category ${\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$ of coherent sheaves is given by

[TABLE]

Denote by ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$ the bounded derived category ${\mathcal{D}}^{b}({\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda}))$ of ${\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$.

For each $\vec{l}\in L_{A}$, set

[TABLE]

where $(R_{A,\Lambda}(\vec{l}))_{\vec{l^{\prime}}}:=(R_{A,\Lambda})_{\vec{l}+\vec{l^{\prime}}}$.

Set $\vec{c}:=a_{1}\vec{x}_{1}=\dots=a_{r}\vec{x}_{r}$. The element $\vec{x}\in L_{A}$ has the unique expression of the form

[TABLE]

We say that $\vec{x}$ is positive if $\vec{x}\neq 0$, $l\geq 0$ and $x_{i}\geq 0$ for $i=1,\dots,r$.

For a ${\mathbb{C}}$-module $M$, set $M^{*}:={\rm Hom}_{{\mathbb{C}}}(M,{\mathbb{C}})$.

Proposition 3.3** (Section 1.8.1 and Section 2.2 in [GL]).**

We have the following:

1. $(\rm{i})$

For $\vec{x},\vec{y}\in L_{A}$ with $\vec{x}-\vec{y}$ positive,

[TABLE] 2. $(\rm{ii})$

Set $\vec{\omega}:=(r-2)\vec{c}-\sum_{i=1}^{r}\vec{x_{i}}\in L_{A}$. For $M_{1},M_{2}\in{\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$, we have the Serre duality isomorphism:

[TABLE] 3. $(\rm{iii})$

The category ${\rm Coh({\mathbb{P}}^{1}_{A,\Lambda})}$ is hereditary, namely, ${\rm Ext}^{i}(M_{1},M_{2})=0$ for $M_{1},M_{2}\in{\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$ if $i\neq 0,1$.

Remark 3.4*.*

It follows from Proposition 3.3 (iii) that each indecomposable object of ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$ is of the form $M[n]$ for some $M\in{\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$ and $n\in{\mathbb{Z}}$.

Proposition 3.5** (Section 1.8.1 and Section 4.1 in [GL]).**

The following sequences are full strongly exceptional collections:

[TABLE]

[TABLE]

In particular, $G:=\displaystyle\bigoplus_{i=1}^{\mu_{A}}E_{i}$ and $G^{*}:=\displaystyle\bigoplus_{i=1}^{\mu_{A}}E^{*}_{i}$ are split-generators of ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$.

It follows from Proposition 3.5 that ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})\cong{\rm per}({\rm End}(G))$ since the triangulated category ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$ is algebraic. Denote by ${\mathcal{N}}({\mathbb{P}}^{1}_{A,\Lambda})$ its numerical Grothendieck group.

Definition 3.6** (Section 2.5 in [GL]).**

Take $[1:\lambda]\in{\mathbb{P}}^{1}\setminus\{\lambda_{1},\dots,\lambda_{r}\}$. Define $S$ and $S_{i,j}$ for $i=1,\dots,r$ and $j=0,\dots,a_{i}-1$ by the following exact sequences:

[TABLE]

[TABLE]

Definition 3.7** (Section 1.8.2 and Section 2.8 in [GL]).**

The rank and degree are homomorphisms defined as follows:

[TABLE]

[TABLE]

Definition 3.8**.**

Denote by ${\rm Pic}({\mathbb{P}}^{1}_{A,\Lambda})$ the group consisting of (isomorphism classes of) indecomposable objects in ${\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$ of rank one with multiplication induced by the tensor product.

Lemma 3.9** (Section 2.1 in [GL]).**

There is an isomorphism of abelian groups

[TABLE]

One of our results is the following Gromov–Yomdin type theorem for an orbifold projective line:

Theorem 3.10**.**

For each auto-equivalence $F$ of ${\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$, we have

[TABLE]

Moreover, $\rho({\mathcal{N}}(F))$ is an algebraic number and $h(F)=0$ if $\chi_{A}\neq 0$.

3.1. Proof of Theorem 3.10 for the case $\chi_{A}\neq 0$

It is important to note that Lenzing–Meltzer ([LM, Proposition 4.2]) shows that, if $\chi_{A}\neq 0$,

[TABLE]

3.1.1. Case $\chi_{A}>0$

Geigle–Lenzing (cf. [GL, Section 5.4.1]) gives an equivalence of triangulated categories

[TABLE]

where $\vec{\Delta}_{A}$ is the extended Dynkin quiver below.

This equivalence with Corollary 2.15 yields $h(F)=\log\rho({\mathcal{N}}(F))$. Then, [MY, Theorem 4.2, Theorem 4.5] show that $\rho({\mathcal{N}}(F))=1$. We have finished the proof. ∎

3.1.2. Case $\chi_{A}<0$

We shall prove that $h(F)=0$ for each $F\in({\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})\ltimes{\rm Pic}({\mathbb{P}}^{1}_{A,\Lambda}))\times{\mathbb{Z}}[1]$ if $\chi_{A}\leq 0$.

Choose $\{[{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}],[S_{1,1}],\dots,[S_{i,j}],\dots,[S_{r,a_{r}-1}],[S]\}$ as a basis of ${\mathcal{N}}({\mathbb{P}}^{1}_{A,\Lambda})$.

Lemma 3.11**.**

For $f\in{\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})$, the automorphism ${\mathcal{N}}(f^{*})$ is a composition of permutations exchanging $[S_{i,j}]$ and $[S_{i^{\prime},j}]$ for $j=1,\dots a_{i-1}$ if $a_{i}=a_{i^{\prime}}$ and fixing $[{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}]$ and $[S]$. In particular, we have $\rho({\mathcal{N}}(f^{*}))=1$.

Proof*.*

This is a direct consequence of [LM, Proposition 3.1]. Note also that $r\geq 3$ since $\chi_{A}\leq 0$ and hence ${\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})$ is a finite group. ∎

Lemma 3.12**.**

For ${\mathcal{L}}\in{\rm Pic}({\mathbb{P}}^{1}_{A,\Lambda})$, we have $\rho({\mathcal{N}}(-\otimes{\mathcal{L}}))=1$.

Proof*.*

We may assume that ${\mathcal{L}}={\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{x}_{i})$. By [GL, (2.5.3) and (2.5.4)], for $i=1,\dots,r$ and $j=1,\dots,a_{i}-1$,

[TABLE]

It follows from the above isomorphisms that the representation matrix of ${\mathcal{N}}(-\otimes{\mathcal{L}})$ in the basis becomes an upper triangular matrix whose diagonal entries are all $1$. Hence, its spectral radius is equal to $1$. ∎

Each auto-equivalence $F$ is represented as $F=f^{*}(-\otimes{\mathcal{L}})[m]$ (cf. (3.16)). Since Lemma 2.7 (v) gives $h(F)=h(f^{*}(-\otimes{\mathcal{L}}))$, we may assume $F=f^{*}(-\otimes{\mathcal{L}})$.

Proposition 3.13**.**

We have

[TABLE]

Proof*.*

Take $G$ and $G^{*}$ as in Proposition 3.5. By Proposition 2.11,

[TABLE]

By straightforward calculation,

[TABLE]

Note that $f^{*}(G^{*})=G^{*}$ and $\deg(f^{*}{\mathcal{L}})=\deg({\mathcal{L}})$ by Lemma 3.11.

Suppose that ${\rm deg}({\mathcal{L}})>0$. For $n\gg 0$ and $\vec{z}\in L_{A}$, we have

[TABLE]

Therefore, Proposition 3.3 (i) and (ii) yield

[TABLE]

Suppose that $\deg({\mathcal{L}})\leq 0$. We choose $\vec{z}\in L_{A}$ so that $\deg({\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{z}))\gg 0$. The elements $G^{\prime}:=G\otimes{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{z})$ and $G^{\prime\prime}:=G^{*}\otimes{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(-\vec{z})$ are also split-generators. Therefore, Proposition 3.3 (i) yields

[TABLE]

Hence it follows from Proposition 3.3 (iii), Lemma 3.11 and Lemma 3.12 that

[TABLE]

∎

To summarize, we have finished the proof of Theorem 3.10 for the case $\chi_{A}<0$

3.2. Proof of Theorem 3.10 for the case $\chi_{A}=0$

Define a homomorphism $\nu:{\mathcal{N}}({\mathbb{P}}^{1}_{A,\Lambda})\rightarrow{\mathbb{Z}}^{2}$ by $\nu([M]):=({\rm rk}([M]),{\rm\deg}([M]))$ and a skew symmetric bilinear form on ${\mathbb{Z}}^{2}$ by

[TABLE]

Lemma 3.14**.**

For $M_{1},M_{2}\in{\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})$, we have

[TABLE]

Proof*.*

It follows from [GL, Section 2.9] with $\chi_{A}=0$. ∎

Lemma 3.14 gives the following natural group homomorphism:

[TABLE]

Denote by ${\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})\subset{\rm Pic}({\mathbb{P}}^{1}_{A,\Lambda})$ the subgroup consisting of elements with degree zero.

Proposition 3.15**.**

There exists the following exact sequence:

[TABLE]

Proof*.*

This is a direct consequence of [LM, Theorem 6.3]. ∎

Lemma 3.16**.**

The map $h:{\rm Auteq}({\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda}))\longrightarrow{\mathbb{R}}_{\geq 0}$, $F\mapsto h(F)$ factors through ${\rm SL}(2,{\mathbb{Z}})$.

Proof*.*

Choose ${\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{z})$ so that $\deg({\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{z}))\gg 0$ and set $G^{\prime}:=G\otimes{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{z})$, $G^{\prime\prime}:=G^{*}\otimes{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(-\vec{z})$. By Lemma 2.7 (v), we can assume that an element $F\in{\rm Auteq}({\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda}))$ is of the form $F=F^{\prime}F_{1}$ with $F^{\prime}\in{\rm Auteq}({\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda}))$ and $F_{1}\in{\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})\ltimes{\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})$. Then there exist $F_{2},\dots,F_{n}\in{\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})\ltimes{\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})$ such that $F^{n}=(F^{\prime}F_{1})^{n}=F^{\prime n}F_{n}\cdots F_{1}$. We have

[TABLE]

and hence,

[TABLE]

The functor $F_{i}$ is of the form $f_{i}^{*}(-\otimes{\mathcal{L}}_{i})$ for some $f_{i}\in{\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})$ and ${\mathcal{L}}_{i}\in{\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})$. For arbitrary $\vec{y}\in L_{A}$, we have $\deg(F_{n}\cdots F_{1}({\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{y})))=\deg({\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{y}))$. Therefore, it follows from Proposition 3.3 (i),(iii) that $\delta^{\prime}_{{\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda})}(G^{\prime},F_{n}\cdots F_{1}G^{\prime\prime})=\left|\chi(G^{\prime},F_{n}\cdots F_{1}G^{\prime\prime})\right|$. Lemma 2.12, Lemma 3.11 and Lemma 3.12 yield

[TABLE]

and hence $h(F)\leq h(F^{\prime})$. We also have $h(F^{\prime})\leq h(F)$ since $F^{\prime}=FF_{1}^{-1}$ and $F_{1}^{-1}$ belongs to ${\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})\ltimes{\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})$. ∎

Proposition 3.17**.**

We have $h(F)=\log\rho({\mathcal{N}}(F))$.

Proof*.*

Since $h(F[1])=h(F)$, we may assume that ${\rm tr}(\varphi(F))\geq 0$. It is easy to calculate $h(F)=0$ if ${\rm tr}(\varphi(F))=0,1$ since $\varphi(F)$ is of finite order and hence $F$ is of finite order up to ${\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})\ltimes{\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})$. If ${\rm tr}(\varphi(F))=2$, then $F=(-\otimes{\mathcal{O}}_{{\mathbb{P}}^{1}_{A,\Lambda}}(\vec{x}))F^{\prime}$ with $F^{\prime}\in{\rm Aut}({\mathbb{P}}^{1}_{A,\Lambda})\ltimes{\rm Pic}^{0}({\mathbb{P}}^{1}_{A,\Lambda})$ for some $\vec{x}\in L_{A}$. It follows from Proposition 3.13 that

[TABLE]

Suppose now that ${\rm tr}(\varphi(F))>2$.

Lemma 3.18**.**

For indecomposable objects $M_{1},M_{2}\in{\rm Coh}({\mathbb{P}}^{1}_{A,\Lambda})$, we have

[TABLE]

Proof*.*

The statement follows from the slope-stability for orbifold projective lines (Proposition 5.2 in [GL]) and Proposition 3.3 (ii). ∎

Lemma 3.19** (Proposition 4.6 in [Kik]).**

Assume that ${\rm tr}(\varphi(F))>2$. There exists a sequence of positive integers ${\bf m}=(m_{2n},\dots,m_{1})$ with $n\geq 1$ such that $\varphi(F)$ is conjugate in ${\rm SL}(2,{\mathbb{Z}})$ to

[TABLE]

For each sequence of positive integers ${\bf m}=(m_{2n},\dots,m_{1})$ with $n\geq 1$, take $F_{\bf m}\in{\rm Auteq}({\mathcal{D}}^{b}({\mathbb{P}}^{1}_{A,\Lambda}))$ so that

[TABLE]

For positive $\vec{x},\vec{y}\in L_{A}$, an elementary calculation gives

[TABLE]

Take $G,G^{*}$ as in Proposition 3.5 and a positive $\vec{x}\in L_{A}$. By Proposition 3.3 (iii), Lemma 3.18, we obtain

[TABLE]

Lemma 3.20**.**

We have

[TABLE]

In particular, $\rho({\mathcal{N}}(F))$ is an algebraic number.

Proof*.*

The inequality $\rho({\mathcal{N}}(F))\geq\rho(\varphi(F))$ follows from the commutativity: $\varphi(F)\circ\nu=\nu\circ{\mathcal{N}}(F)$. The fact that $\varphi$ factors thorough the surjection ${\rm Aut}_{\mathbb{Z}}({\mathcal{N}}({\mathbb{P}}^{1}_{A,\Lambda}),\chi)\to{\rm Aut}_{{\mathbb{Z}}}({\mathbb{Z}}^{2},\chi^{\prime})$ ([LM, Theorem 7.3]) yields the reversed inequality. ∎

Since $\varphi(F_{{\bf m}})$ is conjugate to $\varphi(F)$, it follows from Lemma 3.16 and Lemma 3.20 that

[TABLE]

By Theorem 2.13, we finished the proof of Proposition 3.17, hence of Theorem 3.10. ∎