aa r X i v : . [ m a t h . C T ] D ec MASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY
AKISHI IKEDA
Abstract.
In the pioneer work by Dimitrov-Haiden-Katzarkov-Kontsevich, theyintroduced various categorical analogies from classical theory of dynamical sys-tems. In particular, they defined the entropy of an endofunctor on a triangulatedcategory with a split generator. In the connection between categorical theoryand classical theory, a stability condition on a triangulated category plays therole of a measured foliation so that one can measure the “volume” of objects,called the mass, via the stability condition. The aim of this paper is to establishfundamental properties of the growth rate of mass of objects under the mappingby the endofunctor and to clarify the relationship between the entropy and that.We also show that they coincide under a certain condition. Introduction
In the pioneer work [DHKK14], Dimitrov-Haiden-Katzarkov-Kontsevich intro-duced various categorical analogies of classical theory from dynamical systems. Inparticular, they defined the entropy of an endofunctor on a triangulated categorywith a split generator. One of their motivations comes from the connection betweentheory of stability conditions on triangulated categories and Teichm¨uller theory ofsurfaces [GMN13, BS15]. In this connection, a stability condition on a triangulatedcategory corresponds to a measured foliation (a quadratic differential) on a surface,and the mass of stable objects corresponds to the length of geodesics. Thus themass of objects plays the role of “volume” in some sense. In the work [DHKK14],they also suggested that there is a connection between the growth rate of massof objects under the mapping by an endofunctor and the entropy of that. In thispaper, we establish fundamental properties of the mass growth and clarify the re-lationship between the entropy and that. We also show that they coincide under acertain condition. The result in this paper is motivated by the famous classical work“Volume growth and entropy” by Yomdin [Yom87] on classical dynamical systems.1.1.
Fundamental properties of mass growth.
First we introduce the massgrowth with respect to endofunctors. Let D be a triangulated category and K ( D )be its Grothendieck group. A stability condition σ = ( Z, P ) on D [Bri07] is a pair ofa linear map Z : K ( D ) → C and a family of full subcategories P ( φ ) ⊂ D for φ ∈ R satisfying some axioms (see Definition 2.8). A nonzero object in P ( φ ) is called asemistable object of phase φ . One of the axioms implies that any nonzero object E ∈ D can be decomposed into semistable objects with decreasing phases, i.e. there is a sequence of exact triangles, called a Harder-Narasimhan filtration,0 = E / / E (cid:2) (cid:2) ✆✆✆✆✆✆✆ / / E (cid:2) (cid:2) ✆✆✆✆✆✆✆ / / . . . / / E m − / / E m (cid:1) (cid:1) ✄✄✄✄✄✄✄ A \ \ ✾ ✾ ✾ ✾ A \ \ ✾ ✾ ✾ ✾ A m _ _ ❅ ❅ ❅ ❅ = E with A i ∈ P ( φ i ) and φ > φ > · · · > φ m . Through the Harder-Narasimhanfiltration, the mass of E with a parameter t ∈ R (see Definition 3.1) is defined by m σ,t ( E ) := m X i =1 | Z ( A i ) | e φ i t . Thus a given stability condition defines the “volume” of objects in some sense.Actually in the connection between spaces of stability conditions and Teichm¨ullerspaces, the mass of stable objects gives the length of corresponding geodesics [BS15,GMN13, HKK, Ike]. For an endofunctor F : D → D , we want to consider thegrowth rate of mass of objects under the mapping by F . Therefore we introduce thefollowing quantity. The mass growth with respect to F is the function h σ,t ( F ) : R → [ −∞ , ∞ ] defined by h σ,t ( F ) := sup E ∈D (cid:26) lim sup n →∞ n log( m σ,t ( F n E )) (cid:27) . (As conventions, set m σ,t (0) = 0 and log 0 = −∞ .) Fundamental properties of h σ,t ( F ) are stated as the main result of this paper. We also recall the space ofstability conditions to consider the behavior of h σ,t ( F ) under the deformation of σ .In [Bri07], it was shown that the set of stability conditions Stab( D ) has a naturaltopology and in addition, Stab( D ) becomes a complex manifold.Next we recall the entropy of endofunctors from [DHKK14]. Let D be a tri-angulated category with a split-generator and F : D → D be an endofunctor. In[DHKK14], they introduced the function h t ( F ) : R → [ −∞ , ∞ ), called the entropyof F (see Definition 2.4), and showed various fundamental properties of h t ( F ).In addition, they asked the relationship between the entropy h t ( F ) and the massgrowth h σ,t ( F ) (see [DHKK14, Section 4.5]). Our result is the following. Theorem 1.1 (Theorem 3.5 and Proposition 3.10) . Let D be a triangulated cate-gory, F : D → D be an endofunctor and σ be a stability condition on D . Assume that D has a split-generator G . Then the mass growth h σ,t ( F ) satisfies the followings. (1) If a stability condition τ lies in the same connected component as σ in thespace of stability conditions Stab( D ) , then h σ,t ( F ) = h τ,t ( F ) . ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 3 (2)
The mass growth of the generator G determines h σ,t ( F ) , i.e. h σ,t ( F ) = lim sup n →∞ n log( m σ,t ( F n G )) . (3) An inequality h σ,t ( F ) ≤ h t ( F ) < ∞ holds. In the case t = 0, this result was stated in [DHKK14, Section 4.5] by using thetriangle inequality for mass (see Proposition 3.3). However, the triangle inequalityfor mass is non-trivial even if t = 0 and actually the most technical part in thispaper. Therefore we give a detailed proof of it with a parameter t in Section 3.2.1.2. Lower bound by the spectral radius.
We consider the lower bound of themass growth when t = 0. Since F : D → D preserves exact triangles, F induces alinear map [ F ] : K ( D ) → K ( D ). The spectral radius of [ F ] is defined by ρ ([ F ]) := max {| λ | | λ is an eigenvalue of [ F ] } . Theorem 1.2 (Proposition 3.11) . In the case t = 0 , we have an inequality log ρ ([ F ]) ≤ h σ, ( F ) ≤ h ( F ) for any stability condition σ ∈ Stab( D ) . As known results, if D is saturated, then it was shown in [DHKK14, Theorem2.9] that for a linear map HH ∗ ( F ) : HH ∗ ( D ) → HH ∗ ( D ) induced on the Hochschildhomology of D , the inequality log ρ (HH ∗ ( F )) ≤ h ( F ) holds under some condi-tion for eigenvalues of HH ∗ ( F ). They also conjectured that the inequality holdswithout that condition. Our result Theorem 1.2 holds without any conditions for[ F ], however we use the existence of stability conditions on D . For many examplesin [DHKK14, Kik, KT], it was shown that the equality log ρ ([ F ]) = h ( F ) holds.Kikuta-Takahashi gave a certain conjecture on the equality in [KT, Conjecture 5.3].1.3. Equality between mass growth and entropy.
The remaining importantquestion is to ask when the equality h σ,t ( F ) = h t ( F ) holds. In the following, wegive a sufficient condition for the equality. For a stability condition σ = ( Z, P ),we can associate an abelian category, called the heart of P , as the extension-closedsubcategory generated by objects in P ( φ ) for φ ∈ (0 , P ((0 , σ = ( Z, P ) is called algebraic if the heart P ((0 , AKISHI IKEDA
Theorem 1.3 (Theorem 3.14) . Let G ∈ D be a split-generator and F : D → D be anendofunctor. If a connected component
Stab ◦ ( D ) ⊂ Stab( D ) contains an algebraicstability condition, then for any σ ∈ Stab ◦ ( D ) we have h t ( F ) = h σ,t ( F ) = lim n →∞ n log( m σ,t ( F n G )) . Note that in the above theorem, a stability condition σ is not necessarily analgebraic stability condition.We see a typical example which satisfies the condition in Theorem 1.3 from Sec-tion 4.1. Let A = ⊕ k A k be a dg-algebra such that H ( A ) is a finite dimensionalalgebra and H k ( A ) = 0 for k >
0. Denote by D fd ( A ) the derived category of dg-modules over A with finite dimensional total cohomology, i.e. P k dim H k ( M ) < ∞ .Then there is a bounded t-structure whose heart is isomorphic to the abelian cat-egory of finite dimensional modules over H ( A ). As a result, we can constructalgebraic stability conditions on D fd ( A ). Thus in the context of representationtheory, Theorem 1.3 works well. As an application, we compute the entropy ofspherical twists in Section 4.2.On the other hand, for derived categories coming from algebraic geometry, wecannot find algebraic hearts in general. Only in special cases, for exmaple in thecase that the derived category has a full strong exceptional collection, the workby Bondal [Bon89] enables us to find algebraic hearts. It is an important problemto answer whether the equality h σ,t ( F ) = h t ( F ) holds without the existence ofalgebraic stability conditions.1.4. Categorical theory versus classical theory.
We compere our result withthe famous classical result “Volume growth and entropy” by Yomdin [Yom87]. Let M be a compact smooth manifold and f : M → M be a smooth map. The map f induces a linear map f ∗ : H ∗ ( M ; R ) → H ∗ ( M ; R ) on the homology group H ∗ ( M ; R ).For the map f , we can define the topological entropy h top ( f ) [AKM65] and theinequality log ρ ( f ∗ ) ≤ h top ( f ) was conjectured in [Shu74]. In [Yom87], Yomdinintroduced the volume growth v ( f ) by using a Riemannian metric on M and showedthat log ρ ( f ∗ ) ≤ v ( f ) ≤ h top ( f ) . Our result Theorem 1.2 looks like the categorical analogy of this classical result.On the other hand, the difference between categorical theory and classical theory isthat the categorical entropy h t ( F ) and the mass growth h σ,t ( F ) have the parameter t which measures the growth rate of degree shifts in a triangulated category. Thispoint is an essentially new feature of categorical theory. Acknowledgements.
The author would like to thank Kohei Kikuta, Genki Ouchiand Atsushi Takahashi for valuable discussions and comments.
ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 5
This work is supported by World Premier International Research Center Initia-tive (WPI initiative), MEXT, Japan, JSPS KAKENHI Grant Number JP16K17588and JSPS bilateral Japan-Russia Research Cooperative Program. This paper waswritten while the author was visiting Perimeter Institute for Theoretical Physics byJSPS Program for Advancing Strategic International Networks to Accelerate theCirculation of Talented Researchers. Research at Perimeter Institute is supportedby the Government of Canada through the Department of Innovation, Science andEconomic Development Canada and by the Province of Ontario through the Min-istry of Research, Innovation and Science.
Notations.
We work over a field K . All triangulated categories in this paper are K -linear and their Grothendieck groups are free of finite rank, i.e. K ( D ) ∼ = Z n for some n . An endofunctor F : D → D refers to an exact endofunctor, i.e. F preserves allexact triangles and commutes with degree shifts. The natural logarithm is extendedto log : [0 , ∞ ) → [ −∞ , ∞ ) by setting log 0 := −∞ .2. Preliminaries
In this section, we prepare basic terminologies mainly from [DHKK14, Bri07].2.1.
Complexity and entropy.
First we recall the notion of complexity and en-tropy from [DHKK14, Section 2].Let D be a triangulated category. A triangulated subcategory is called thick if it isclosed under taking direct summands. For an object E ∈ D , we denote by h E i ⊂ D the smallest thick triangulated subcategory containing E . An object G ∈ D is calleda split-generator if h G i = D . This implies that for any object E ∈ D , there is someobject E ′ ∈ D such that we have a sequence of exact triangles0 = A / / A (cid:0) (cid:0) ✁✁✁✁✁✁✁ / / A (cid:0) (cid:0) ✁✁✁✁✁✁✁ / / . . . / / A k − / / A k (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) G [ n ] ^ ^ ❂ ❂ ❂ ❂ G [ n ] ^ ^ ❂ ❂ ❂ ❂ G [ n k ] ` ` ❇ ❇ ❇ ❇ = E ⊕ E ′ with n i ∈ Z . We note that the object E ′ and the above sequence are not unique. Definition 2.1 ([DHKK14], Definition 2.1) . Let E and E be objects in D . The complexity of E relative to E is the function δ t ( E , E ) : R → [0 , ∞ ) defined by δ t ( E , E ) := E ∼ = 0inf k X i =1 e n i t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A . . . A k − E ⊕ E ′ E [ n ] . . . E [ n k ] / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ _ _ ❄ ❄ ❄ / / (cid:127) (cid:127) ⑧⑧⑧⑧⑧ _ _ ❄ ❄ ❄ if E ∈ h E i∞ if E / ∈ h E i . By definition, we have an inequality 0 < δ t ( G, E ) < ∞ for a split-generator G ∈ D and a nonzero object E ∈ D . We recall fundamental inequalities for complexity. AKISHI IKEDA
Proposition 2.2 ([DHKK14], Proposition 2.3) . For E , E , E ∈ D , (1) δ t ( E , E ) ≤ δ t ( E , E ) δ t ( E , E ) , (2) δ t ( E , E ⊕ E ) ≤ δ t ( E , E ) + δ t ( E , E ) , (3) δ t ( F ( E ) , F ( E )) ≤ δ t ( E , E ) for an endofunctor D → D . Similar to [DHKK14, Proposition 2.3], it is easy to check the following.
Lemma 2.3.
For objects
D, E , E , E ∈ D , if there is an exact triangle E → E → E → E [1] , then δ t ( D, E ) ≤ δ t ( D, E ) + δ t ( D, E ) . Now we introduce the notion of the entropy of endofunctors. The entropy of anendofunctor F measures the growth rate of complexity δ t ( G, F n G ) as n → ∞ . Definition 2.4 ([DHKK14], Definition 2.5) . Let D be a triangulated category witha split-generator G and let F : D → D be an endofunctor. The entropy of F is thefunction h t ( F ) : R → [ −∞ , ∞ ) defined by h t ( F ) := lim n →∞ n log δ t ( G, F n G ) . By [DHKK14, Lemma 2.6], it follows that h t ( F ) is well-defined and h t ( F ) < ∞ .2.2. Bounded t-structures and the associated cohomology.Definition 2.5 ([BBD82]) . A t-structure on D is a full subcategory F ⊂ D satis-fying the following conditions:(a) F [1] ⊂ F ,(b) define F ⊥ := { F ∈ D| Hom(
D, F ) = 0 for all D ∈ F } , then for every object E ∈ D there is an exact triangle D → E → F → D [1] in D with D ∈ F and F ∈ F ⊥ .In addition, the t-structure F ⊂ D is said to be bounded if F satisfies the condition D = [ i,j ∈ Z F ⊥ [ i ] ∩ F [ j ] . For a t-structure
F ⊂ D , we define the heart
H ⊂ D by H := F ⊥ [1] ∩ F . It was proved in [BBD82] that H becomes an abelian category. Bridgeland gave thecharacterization of the heart of a bounded t-structure as follows. Lemma 2.6 ([Bri07], Lemma 3.2) . Let
H ⊂ D be a full additive subcategory. Then H is the heart of a bounded t-structure if and only if the following conditions hold: (a) if k > k ∈ Z and A i ∈ H [ k i ] ( i = 1 , , then Hom D ( A , A ) = 0 , ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 7 (b) for = E ∈ D , there is a finite sequence of integers k > k > · · · > k m and a sequence of exact triangles E / / E (cid:2) (cid:2) ✆✆✆✆✆✆✆ / / E (cid:2) (cid:2) ✆✆✆✆✆✆✆ / / . . . / / E m − / / E m (cid:1) (cid:1) ✄✄✄✄✄✄✄ A \ \ ✾ ✾ ✾ ✾ A \ \ ✾ ✾ ✾ ✾ A m _ _ ❅ ❅ ❅ ❅ = E with A i ∈ H [ k i ] for all i . The above filtration in the condition (b) defines the k -th cohomology H k ( E ) ∈ H of the object E by H k ( E ) := ( A i [ − k i ] if k = − k i . This cohomology becomes a cohomological functor from D to H , i.e. if there is anexact triangle D → E → F → E [1], then we can obtain a long exact sequence · · · → H k − ( F ) → H k ( D ) → H k ( E ) → H k ( F ) → H k +1 ( D ) → · · · in the abelian category H . In the last of this section, we introduce the special classof bounded t-structures. Definition 2.7.
We say that the heart of a bounded t-structure is algebraic if itis a finite length abelian category with finitely many isomorphism classes of simpleobjects.If D has an algebraic heart H with simple objects S , . . . , S n , then it is easy tosee that the direct sum G := ⊕ ni =1 S i becomes a split-generator of D .2.3. Bridgeland stability conditions.
In [Bri07], Bridgeland introduced the no-tion of a stability condition on a triangulated category as follows.
Definition 2.8.
Let D be a triangulated category and K ( D ) be its Grothendieckgroup. A stability condition σ = ( Z, P ) on D consists of a group homomorphism Z : K ( D ) → C , called a central charge , and a family of full additive subcategories P ( φ ) ⊂ D for φ ∈ R satisfying the following conditions:(a) if 0 = E ∈ P ( φ ), then Z ( E ) = m ( E ) exp( iπφ ) for some m ( E ) ∈ R > ,(b) for all φ ∈ R , P ( φ + 1) = P ( φ )[1],(c) if φ > φ and A i ∈ P ( φ i ) ( i = 1 , D ( A , A ) = 0,(d) for 0 = E ∈ D , there is a finite sequence of real numbers φ > φ > · · · > φ m AKISHI IKEDA and a sequence of exact triangles0 = E / / E (cid:2) (cid:2) ✆✆✆✆✆✆✆ / / E (cid:2) (cid:2) ✆✆✆✆✆✆✆ / / . . . / / E m − / / E m (cid:1) (cid:1) ✄✄✄✄✄✄✄ A \ \ ✾ ✾ ✾ ✾ A \ \ ✾ ✾ ✾ ✾ A m _ _ ❅ ❅ ❅ ❅ = E with A i ∈ P ( φ i ) for all i .We write φ + σ ( E ) := φ and φ − σ ( E ) := φ m . Nonzero objects in P ( φ ) are called σ -semistable of phase φ in σ . The sequence of exact triangles in (d) is called a Harder-Narasimhan filtration of E with semistable factors A , . . . , A m of phases φ > · · · > φ m .In addition to the above axioms, we always assume that our stability conditionshave the support property in [KS]. Let k · k be some norm on K ( D ) ⊗ R . A stabilitycondition σ = ( Z, P ) satisfies the support property if there is a some constant C > | Z ( E ) |k [ E ] k > C for all σ -semistable objects E ∈ D .For an interval I ⊂ R , we denote by P ( I ) the extension-closed subcategorygenerated by objects in P ( φ ) for φ ∈ I , namely P ( I ) := { E ∈ D | φ ± σ ( E ) ∈ I } ∪ { } . From a stability condition ( Z, P ), we can construct a bonded t-structure F := P ((0 , ∞ )) and its heart is given by H = P ((0 , Algebraic stability conditions.
In [Bri07], Bridgeland gave the alternativedescription of a stability condition on D as a pair of a bounded t-structure anda central charge on its heart. By using this description, we construct algebraicstability conditions. Definition 2.9.
Let H be an abelian category and let K ( H ) be its Grothendieckgroup. A central charge on H is a group homomorphism Z : K ( H ) → C such thatfor any nonzero object 0 = E ∈ H , the complex number Z ( E ) lies in the semi-closedupper half-plane H := { re iπφ ∈ C | r ∈ R > , φ ∈ (0 , } .For any nonzero object E ∈ H , define the phase of E by φ ( E ) := 1 π arg Z ( E ) ∈ (0 , . An object 0 = E ∈ H is called Z -semistable if every subobject 0 = A ⊂ E satisfies φ ( A ) ≤ φ ( E ). A Harder-Narasimhan filtration of 0 = E ∈ H is the filtration0 = E ⊂ E ⊂ · · · ⊂ E m − ⊂ E m = E ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 9 whose extension factors F i := E i /E i − are Z -semistable with decreasing phases φ ( F ) > · · · > φ ( F m ) . A central charge Z is said to have the Harder-Narasimhan property if any nonzeroobject of H has a Harder-Narasimhan filtration. The following gives the anotherdefinition of a stability condition. Proposition 2.10 ([Bri07], Proposition 5.3) . Giving a stability condition on D isequivalent to giving a heart H of a bounded structure on D and a central charge on H with the Harder-Narasimhan property. In Proposition 2.10, the pair ( Z, H ) is constructed from a stability condition( Z, P ) by setting H := P ((0 , Definition 2.11.
A stability condition ( Z, P ) is called algebraic if the correspondingheart H = P ((0 , H ⊂ D be an algebraic heart with simple objects S , . . . , S n . Then the Grothendieckgroup is given by K ( H ) ∼ = ⊕ ni =1 Z [ S i ]. Take ( z , . . . , z n ) ∈ H n and define the cen-tral charge Z : K ( H ) → C by the linear extension of Z ( S i ) := z i . Then Z has theHarder-Narasimhan property by [Bri07, Proposition 2.4]. Thus ( Z, H ) becomes astability condition on D .2.5. Harder-Narasimhan polygons.
In this section, we discuss the Harder-Narasimhanpolygon following [Bay]. This plays a key role to show the triangle inequality formass in Section 3.2. The following is based on [Bay, Section 3].
Definition 2.12.
Let H be an abelian category and Z be a central charge on it.For an object E ∈ H , the Harder-Narasimhan polygon HN Z ( E ) of E is the convexhull of the subset { Z ( A ) ∈ C | A ⊂ E } ⊂ C in the complex plane.It is clear from the definition that if F ⊂ E , then HN Z ( F ) ⊂ HN Z ( E ). TheHarder-Narasimhan polygon HN Z ( E ) is called polyhedral on the left if it has finitelymany extremal points 0 = z , z , . . . , z k = Z ( E ) such that HN Z ( E ) lies to the rightof the path z z . . . z k . This implies that the intersection of HN Z ( E ) and the closedhalf-plane to the left of the line through 0 and Z ( E ) becomes a polygon with vertices z , z , . . . , z k (see Figure 1). Proposition 2.13 ([Bay], Proposition 3.3) . The object E has a Harder-Narasimhanfiltration if and only if HN Z ( E ) is polyhedral on the left. In particular, if the Harder-Narasimhan filtration of E is given by E ⊂ E ⊂ E ⊂ · · · E k = E, z z z z z Figure 1.
Harder-Narasimhan polygon. then extremal points of HN Z ( E ) are given by z i = Z ( E i ) for i = 0 , , . . . , k . Topology on the space of stability conditions.
In [Bri07], Bridgelandintroduced a natural topology on the space of stability conditions and showed thatthis space becomes a complex manifold. In the following, we recall his construction.Let Stab( D ) be the set of stability conditions on a triangulated category D with thesupport property. For stability conditions σ = ( Z, P ) and τ = ( W, Q ) in Stab( D ),set d ( P , Q ) := sup = E ∈D (cid:8) | φ − σ ( E ) − φ − τ ( E ) | , | φ + σ ( E ) − φ + τ ( E ) | (cid:9) ∈ [0 , ∞ ] . and k Z − W k σ := sup (cid:26) | Z ( E ) − W ( E ) || Z ( E ) | (cid:12)(cid:12)(cid:12)(cid:12) E is σ -semistable (cid:27) ∈ [0 , ∞ ] . Define a subset B ǫ ( σ ) ⊂ Stab( D ) by B ǫ ( σ ) := { τ = ( W, Q ) ∈ Stab( D ) | d ( P , Q ) < ǫ, k Z − W k σ < sin( πǫ ) } for 0 < ǫ < .In [Bri07, Section 6], it was shown that a family of subsets (cid:26) B ǫ ( σ ) ⊂ Stab( D ) (cid:12)(cid:12)(cid:12)(cid:12) σ ∈ Stab( D ) , < ǫ < (cid:27) becomes an open basis of a topology on Stab( D ). In [Bri07], Bridgeland showed acrucial theorem. Theorem 2.14 ([Bri07], Theorem 1.2) . The projection map of central charges π : Stab( D ) −→ Hom Z ( K ( D ) , C ) , ( Z, P ) Z is a local isomorphism of topological spaces. In particular, π induces a complexstructure on Stab( D ) . ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 11 Mass growth of objects and categorical entropy
Mass with a parameter and complexity.
In this sectioin, we introducethe mass growth of objects and show fundamental properties of it.
Definition 3.1 ([DHKK14], Section 4.5) . Take a stability condition σ = ( Z, P ) on D . Let E ∈ D be a nonzero object with semistable factors A , . . . , A m of phases φ > · · · > φ m . The mass of E with a parameter t ∈ R is the function m σ,t ( E ) : R → R > defined by m σ,t ( E ) := m X i =1 | Z ( A i ) | e φ i t . When t = 0, m σ, ( E ) is called the mass of E and simply written as m σ ( E ) := m σ, ( E ). As a convention, set m σ,t ( E ) := 0 if E ∼ = 0.In the following, if σ is clear in the context, we often drop it from the notationand write m t ( E ). Similar to the growth rate of complexity of a generator withrespect to endofunctors, we consider the growth rate of mass of objects. Definition 3.2 ([DHKK14], Section 4.5) . Let σ be a stability condition on D and F : D → D be an endofunctor. The mass growth with respect to F is the function h σ,t ( F ) : R → [ −∞ , ∞ ] defined by h σ,t ( F ) := sup E ∈D (cid:26) lim sup n →∞ n log( m σ,t ( F n E )) (cid:27) . In the rest of this section, we study fundamental properties of h σ,t ( F ). Thetriangle inequality for m σ,t plays an important role. Proposition 3.3.
For objects
D, E, F ∈ D , if there is an exact triangle D → E → F → D [1] , then m σ,t ( E ) ≤ m σ,t ( D ) + m σ,t ( F ) . The proof of Proposition 3.3 is given in Section 3.2.
Proposition 3.4.
Let σ be a stability condition on D . Then m σ,t ( E ) ≤ m σ,t ( D ) δ t ( D, E ) for any objects D, E ∈ D . Proof.
It is sufficient to show the case E ∈ h D i . Then by the definition ofcomplexity δ t ( D, E ), for any ǫ > / / A (cid:0) (cid:0) ✁✁✁✁✁✁✁ / / A (cid:0) (cid:0) ✁✁✁✁✁✁✁ / / . . . / / A k − / / E ⊕ E ′ | | ③③③③③③③③③ D [ n ] ] ] ✿ ✿ ✿ ✿ D [ n ] ^ ^ ❂ ❂ ❂ ❂ D [ n k ] ` ` ❇ ❇ ❇ ❇ such that k X i =1 e n i t < δ t ( D, E ) + ǫ. Note that m σ,t satisfies m σ,t ( D [ n ]) = m σ,t ( D ) · e nt for D ∈ D and n ∈ Z . By usingthe inequality in Proposition 3.3 repeatedly, we have m σ,t ( E ) ≤ m σ,t ( E ⊕ E ′ ) ≤ k X i =1 m σ,t ( D [ n i ]) ≤ m σ,t ( D ) · k X i =1 e n i t ! ≤ m σ,t ( D ) δ t ( D, E ) + ǫ · m σ,t ( D )for any ǫ >
0. This implies the result. (cid:3)
Now we show fundamental properties of the mass growth.
Theorem 3.5.
Let F : D → D be an endofunctor and σ be a stability conditionon D . Assume that D has a split-generator G ∈ D . Then the mass growth h σ,t ( F ) satisfies the followings. (1) h σ,t ( F ) is given by h σ,t ( F ) = lim sup n →∞ n log( m σ,t ( F n G )) . (2) We have an inequality h σ,t ( F ) ≤ h t ( F ) < ∞ where h t ( F ) is the entropy of F (see Definition 2.4). Proof.
By Proposition 2.2 (3) and Proposition 3.4, we have m t ( F n E ) ≤ m t ( F n G ) δ t ( F n G, F n E ) ≤ m t ( F n G ) δ t ( G, E )for any object E ∈ D . Hencelim sup n →∞ n log m t ( F n E ) ≤ lim sup n →∞ n log m t ( F n G )and this inequality implies (1). Again by Proposition 3.4, we have m t ( F n G ) ≤ m t ( G ) δ t ( G, F n G ) . Hence lim sup n →∞ n log m t ( F n G ) ≤ lim n →∞ n log δ t ( G, F n G )and this inequality implies (2). (cid:3) ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 13
Triangle inequality for mass with a parameter.
We prove Proposition3.3. Recall the notation H = { re iπφ | r > , φ ∈ (0 , } . For a complex number z ∈ H , define the function of t ∈ R by g t ( z ) := | z | e φ ( z ) t where φ ( z ) is the phase of z given by φ ( z ) := (1 /π ) arg z ∈ (0 , g t ( z ). Lemma 3.6.
For z , z ∈ H , an inequality g t ( z + z ) ≤ g t ( z ) + g t ( z ) holds. z z z + z πaπb Figure 2.
Triangle consisting of vertices 0 , z , z + z . Proof.
Set φ := φ ( z ) , φ := φ ( z ) and φ := φ ( z + z ). If φ = φ , the resultis trivial. We consider the case φ > φ . By applying the law of sine for the triangleconsisting of vertices 0 , z , z + z (see Figure 2), we obtain | z + z | = d sin( πa + πb ) , | z | = d sin πb, | z | = d sin πa where a = φ − φ , b = φ − φ and d is the diameter of the excircle of the triangle.By inputting these parameters, the inequality g t ( z + z ) ≤ g t ( z ) + g t ( z ) becomessin( πa + πb ) ≤ e at sin πb + e − bt sin πa where 0 < a, b <
1. Dividing by sin πa sin πb and applying the addition formula, wehave e at − cos πa sin πa + e − bt − cos πb sin πb ≥ . After setting c = − b , the above inequality is equivalent to f ( a ) ≥ f ( c ) for − 0) and (0 , f ( x ) at the zero is given by lim x →± f ( x ) = tπ . (cid:3) The triangle inequality for g t ( z ) implies the following. Lemma 3.7. Let z , . . . , z k and w , . . . , w l be complex numbers in H with z k = w l and set z = w = 0 . If they satisfy the following conditions (see the left of Figure3): (a) φ ( z i − z i − ) > φ ( z i +1 − z i ) and φ ( w j − w j − ) > φ ( w j +1 − w j ) for i = 1 , . . . , k and j = 1 , . . . , l , (b) the polygon w w w . . . w l w contains the polygon z z z . . . z k z ,then k X i =1 g t ( z i − z i − ) ≤ l X j =1 g t ( w j − w j − ) . z z z z z = w w w w Figure 3. Polygons and a triangulation of the encircled domain. Proof. By the condition ( b ), there is a unique domain encircled by two paths z z z . . . z k and w w w . . . w l . By the convexity condition ( a ), we can triangulatethis domain as in the right of Figure 3. Applying the triangle inequality for g t ( z )(Lemma 3.2) repeatedly, we obtain the result. (cid:3) Lemma 3.8. Let σ = ( Z, P ) be a stability condition and H = P ((0 , be theassociated heart. If there is a short exact sequence → A → B → C → in H and C ∈ P (1) , then m t ( A ) ≤ m t ( B ) + e − t m t ( C ) . Proof. Let 0 = A ⊂ A ⊂ A ⊂ · · · ⊂ A l = A B ⊂ B ⊂ B ⊂ · · · ⊂ B k − = B ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 15 be Harder-Narasimhan filtrations of A and B . Set z i := Z ( A i ) for i = 0 , , . . . , k , w j := Z ( B j ) for j = 0 , , . . . , l − w l := Z ( B ) − Z ( C ) = Z ( A ). Thenby definition of the Harder-Narasimhan filtration, these complex numbers sat-isfy the condition ( a ) in Lemma 3.7. Consider the Harder-Narasimhan polygonsHN Z ( A ) and HN Z ( B ) (see Definition 2.12). By Proposition 2.13, complex num-bers z , z , . . . , z k and w , w , . . . , w l − are extremal points of HN Z ( A ) and HN Z ( B )respectively. Thus the intersection of HN Z ( A ) and the left of the line through 0and z k = Z ( A ) is the polygon z z z . . . z k z and the intersection of HN Z ( B ) andthe left of the line through 0 and w l = Z ( A ) is the polygon w w w . . . w l w .Since HN Z ( A ) ⊂ HN Z ( B ), the polygon w w w . . . w l w contains the polygon z z z . . . z k z and this implies the condition ( b ) in Lemma 3.7. Since m t ( A ) = k X i =1 g t ( z i − z i − ) , m t ( B ) = l − X j =1 g t ( w j − w j − ) , e − t m t ( C ) = g t ( w l − w l − ) , applying Lemma 3.7, we obtain the result. (cid:3) Proof of Proposition 3.3. Assume that there is a exact triangle D → E → F → D [1]. From a Harder-Narasimhan filtration of E , we can construct the dualHarder-Narasimhan filtration D = D m / / D m − (cid:127) (cid:127) ⑦ ⑦ ⑦ ⑦ / / D m − } } ④ ④ ④ ④ / / . . . / / D / / D (cid:2) (cid:2) ✆ ✆ ✆ ✆ A m ] ] ❀❀❀❀❀❀❀ A m − a a ❈❈❈❈❈❈❈❈ A \ \ ✾✾✾✾✾✾✾ = 0with A i ∈ P ( φ i ) and φ m > φ m − > · · · φ . Applying the octahedra axiom for theabove sequence together with the exact triangle D → E → F → D [1], we canconstruct a sequence of exact triangles E = E m / / E m − (cid:127) (cid:127) ⑧ ⑧ ⑧ ⑧ / / E m − } } ④ ④ ④ ④ / / . . . / / E / / F (cid:3) (cid:3) ✞ ✞ ✞ ✞ A m ] ] ❀❀❀❀❀❀❀ A m − a a ❈❈❈❈❈❈❈❈ A \ \ ✽✽✽✽✽✽✽ . Since m t ( D ) = P mi =1 | Z ( A i ) | e tφ i and A i is semistable, the problem is reduced to thecase that D is semistable. Without loss of generality we can assume D ∈ P (1). Bytaking the cohomology associated with the heart H = P ((0 , → H − ( E ) → H − ( F ) → H ( D ) → H ( E ) → H ( F ) → H i ( E ) ∼ = H i ( F ) for i = − , H . If 1 > φ + ( H ( E )), thenthe map f : H ( D ) → H ( E ) is zero. Hence the long exact sequence splits into → H − ( E ) → H − ( F ) → H ( D ) → H ( E ) ∼ = H ( F ). From Lemma 3.8,we have m t ( H − ( E )) e t ≤ m t ( H − ( F )) e t + m t ( D ) . Thus we obtain the result. If the maps f : H ( D ) → H ( E ) is not zero, then thelong exact sequence splits into two short exact sequences0 → H − ( E ) → H − ( F ) → Ker f → → Im f → H ( E ) → H ( F ) → . Let E + ∈ P (1) the semistable factor of E with the phase one. Note that m t ( D ) = m t (Ker f ) + m t (Im f ) since Ker f ⊂ D ∈ P (1) and Im f ⊂ E + ∈ P (1). Again byLemma 3.8, we have m t ( H − ( E )) e t ≤ m t ( H − ( F )) e t + m t (Ker f )and it is easy to check that m t ( H ( E )) = m t (Im f ) + m t ( H ( F )). (cid:3) Mass growth and deformation of stability conditions. The aim of thissection is to show that for a stability condition σ and an endofunctor F , the massgrowth h σ,t ( F ) is stable under the continuous deformation of σ . The followinginequality was shown in [Bri07, Proposition 8.1] when t = 0. Proposition 3.9. Let σ = ( Z, P ) ∈ Stab( D ) be a stability condition on D . If τ =( W, Q ) ∈ B ǫ ( σ ) with small enough ǫ > , then there are functions C , C : R → R > such that C ( t ) m τ,t ( E ) < m σ,t ( E ) < C ( t ) m τ,t ( E ) for all = E ∈ D . Proof. We use the argument similar to the proof of [Bri07, Proposition 8.1]. Itis sufficient to show that for τ = ( W, Q ) ∈ B ǫ ( σ ) with small enough ǫ > 0, there issome constants C > r > m τ,t ( E ) < Ce r | t | m σ,t ( E )for any nonzero object E ∈ D . We first consider the case φ + σ ( E ) − φ − σ ( E ) < η for0 < η < . In this case, it was shown in the proof of [Bri07, Proposition 8.1] thatthere is a constant C ( ǫ, η ) > m τ ( E ) ≤ C ( ǫ, η ) m σ ( E )and C ( ǫ, η ) → { ǫ, η } → 0. Note that φ + σ ( E ) − φ − σ ( E ) < η implies φ ± σ ( E ) ∈ ( ψ, ψ + η ) for some ψ ∈ R . Since d ( P , Q ) < ǫ , we have φ ± τ ( E ) ∈ ( ψ − ǫ, ψ + ǫ + η ).By definition of m σ,t ( E ) and m τ,t ( E ), it follows that m τ,t ( E ) ≤ m τ ( E ) exp (cid:0) φ + τ ( E ) | t | (cid:1) , m σ ( E ) exp (cid:0) φ − σ ( E ) | t | (cid:1) ≤ m σ,t ( E ) . ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 17 Since ψ < φ − σ ( E ) and φ + τ ( E ) < ψ + ǫ + η , we have an inequality m τ,t ( E ) ≤ C ( ǫ, η ) e ( ǫ + η ) | t | m σ,t ( E ) . Next we consider a general nonzero object E . Take a real number φ and a positiveinteger n . For k ∈ Z , define intervals I k := [ φ + knǫ, φ + ( k + 1) nǫ ) , J k := [ φ + knǫ − ǫ, φ + ( k + 1) nǫ + ǫ )and let α k and β k be the truncation functors projecting into the subcategories Q ( I k )and P ( J k ) respectively. Again by the argument in the proof of [Bri07, Proposition8.1], for small enough nǫ , we have m τ,t ( E ) = X k m τ,t ( α k ( E )) ≤ X k m τ,t ( β k ( E )) < C ( ǫ, ( n +2) ǫ ) e ( n +3) ǫ | t | X k m σ,t ( β k ( E )) . On the other hand, we can choose φ so that X k m σ,t ( β k ( E )) ≤ n + 2 n m σ,t ( E ) . By taking the limits ǫ → n → ∞ in keeping with nǫ → 0, the result follows. (cid:3) From Proposition 3.9, we immediately have the following. Proposition 3.10. Let F : D → D be an endofunctor, and σ and τ be stabilityconditions on D . If σ and τ lie in the same connected component in Stab( D ) , then h σ,t ( F ) = h τ,t ( F ) . Proof. Let σ, τ ∈ Stab( D ) be stability conditions such that τ ∈ B ǫ ( σ ) for smallenough ǫ > 0. Then Proposition 3.9 implies h σ,t ( F ) = h τ,t ( F ). Thus h σ,t ( F ) islocally constant on the topological space Stab( D ). (cid:3) Lower bound of the mass growth by the spectral radius. Let F : D → D be an endofunctor. Since F preserves exact triangles in D , F induces a linear map[ F ] : K ( D ) → K ( D ) . The spectral radius of [ F ] is defined by ρ ([ F ]) := max {| λ | | λ is an eigenvalue of [ F ] } . Proposition 3.11. For any stability condition σ ∈ Stab( D ) , we have an inequality log ρ ([ F ]) ≤ h σ, ( F ) . Proof. Set K ( D ) C := K ( D ) ⊗ C . Let A , . . . , A n ∈ D be objects whose classes[ A ] , . . . , [ A n ] form a basis of K ( D ) C . Take an eigenvector v = n X i =1 a i [ A i ] ∈ K ( D ) C ( a i ∈ C ) for the eigenvalue λ ∈ C of [ F ] satisfying | λ | = ρ ([ F ]). First we consider the casethat a stability condition σ = ( Z, P ) satisfies Z ( v ) = 0. Note that the mass satisfies | Z ( E ) | ≤ m σ ( E ) and m σ ( E ⊕ E ′ ) = m σ ( E ) + m σ ( E ′ ) for any objects E, E ′ ∈ D .Then | λ | k | Z ( v ) | = | Z ( λ k v ) | = | Z ([ F ] k v ) | ≤ n X i =1 | a i | · | Z ( F k A i ) |≤ n X i =1 l i m σ ( F k A i ) = m σ (cid:16) F k (cid:16) ⊕ ni =1 A ⊕ l i i (cid:17)(cid:17) where l , . . . , l n are positive integers satisfying | a i | ≤ l i . Since | Z ( v ) | > 0, we havelog ρ ([ F ]) = lim k →∞ k log( | λ | k | Z ( v ) | ) ≤ lim sup k →∞ k log( m σ ( F k E )) ≤ h σ, ( F )where E = ⊕ ni =1 A ⊕ l i i . Next consider the case Z ( v ) = 0. Then by Theorem 2.14,we can deform σ = ( Z, P ) to σ ′ = ( Z ′ , P ′ ) so that Z ′ ( v ) = 0. Again we havelog ρ ([ F ]) ≤ h σ ′ , ( F ) and Proposition 3.10 implies h σ, ( F ) = h σ ′ , ( F ). (cid:3) Mass growth via algebraic stability conditions. If a triangulated cate-gory has an algebraic stability condition, then we can show that the mass growthcoincides with the entropy. Let H ⊂ D be an algebraic heart with simple objects S , . . . , S n . Then the Grothendieck group is given by K ( D ) ∼ = n M i =1 Z [ S i ] . The class of an object E ∈ H is written as [ E ] = P ni =1 d i [ S i ] with d i ∈ Z ≥ . Wedefine the dimension of E by dim E := P ni =1 d i ∈ Z ≥ . Then the dimension givesthe upper bound of the complexity for objects in H . Lemma 3.12. Let H ⊂ D be an algebraic heart with simple objects S , . . . , S n .Then for the split-generator G := ⊕ ni =1 S i , we have an inequality δ t ( G, E ) ≤ dim E. Proof. Since H is a finite length abelian category, for any object E ∈ H thereis a Jordan-H¨older filtration0 = E ⊂ E ⊂ E ⊂ · · · ⊂ E l = E of length l = dim E with E i /E i − ∈ { S , . . . , S n } . As a result, we can construct afiltration 0 = E ′ ⊂ E ′ ⊂ E ′ ⊂ · · · ⊂ E ′ l = E ⊕ E ′ of length l = dim E with E ′ i /E ′ i − = G and this implies the result. (cid:3) ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 19 Following Section 2.4, we construct the special algebraic stability condition. Foran algebraic heart H ⊂ D with simple objects S , . . . , S n , define the central charge Z : K ( D ) ∼ = n M i =1 Z [ S i ] → C by Z ( S i ) := i . Then the pair σ := ( Z , H ) becomes an algebraic stability condi-tion. By definition, the mass of an object E ∈ H is given by m σ ,t ( E ) = dim E · e t . Together with Lemma 3.12, we obtain the following inequality. Proposition 3.13. For the generator G = ⊕ ni =1 S i and the algebraic stability con-dition σ = ( Z , H ) , we have an inequality δ t ( G, E ) ≤ e − t m σ ,t ( E ) . Proof. For an object E ∈ D , we denote by H k ( E ) ∈ H the cohomologyassociated with the heart H (see Section 2.2). By using Lemma 2.3 and Lemma3.12, we have δ t ( G, E ) ≤ X k δ t ( G, H k ( E )) e − kt ≤ X k dim H k ( E ) e − kt . On the other hand, the definition of m σ ,t ( E ) implies m σ ,t ( E ) = X k m σ ,t ( H k ( E )) e − kt = X k dim H k ( E ) e t e − kt . Thus we obtain the result. (cid:3) We show the main result of this section. Theorem 3.14. Let G ∈ D be a split-generator and F : D → D be an endofunc-tor. If a connected component Stab ◦ ( D ) ⊂ Stab( D ) contains an algebraic stabilitycondition, then for any σ ∈ Stab ◦ ( D ) we have h t ( F ) = h σ,t ( F ) = lim n →∞ n log( m σ,t ( F n G )) . Proof. Let H be an algebraic heart with simple objects S , . . . , S n and set G = ⊕ ni =1 S i . Consider the special algebraic stability condition σ = ( Z , H ) whichis constructed in this section. By Proposition 3.10, it is sufficient to show that h σ ,t ( F ) = h t ( F ) . By [DHKK14, Lemma 2.6], the limit h t ( F ) = lim n →∞ n log δ t ( G, F n G ) converges. On the other hand, by Proposition 3.4 and Proposition 3.13, we have e t δ t ( G, F n G ) ≤ m σ ,t ( F n G ) ≤ m σ ,t ( G ) δ t ( G, F n G ) . Hence the limit lim n →∞ n log( m σ ,t ( F n G ))converges and coincides with h t ( F ). (cid:3) Applications Entropy on the derived categories of non-positive dg-algebras. In thissection, we discuss the entropy of endofunctors on the derived categories of non-positive dg-algebras. In this case, we can describe the entropy as the growth rateof the Hilbert-Poincar´e polynomial of a generator.Let A = ⊕ k ∈ Z A k be a dg-algebra over K satisfying the following conditions:(a) H k ( A ) = 0 for i > H ( A ) is a finite dimensional algebra over K .Let D ( A ) be the derived category of dg-modules over A and D fd ( A ) be thefull subcategory of D ( A ) consisting of dg-modules with finite dimensional totalcohomology, i.e. D fd ( A ) := ( M ∈ D ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k dim H k ( M ) < ∞ ) . Define the full subcategory F ⊂ D fd ( A ) by F := { M ∈ D fd ( A ) | H k ( M ) = 0 for k > } . Then F becomes a bounded t-structure on D fd ( A ). The heart H s of F is calledthe standard heart . It is known that the 0-th cohomology functor H : D fd ( A ) → mod- H ( A ) induces an equivalence of abelian categories: H : H s ∼ −→ mod- H ( A )where mod- H ( A ) is an abelian category of finite dimensional H ( A )-modules. (Fordetails, see [Ami09, Section 2].) Since H ( A ) is a finite dimensional algebra, H s becomes an algebraic heart. Thus we can construct an algebraic stability conditionon D fd ( A ). Applying Theorem 3.14, we obtain the following. Proposition 4.1. Let Stab ◦ ( D fd ( A )) be the connected component which containsstability conditions with the standard heart H s . Then for any stability conditions σ ∈ Stab ◦ ( D fd ( A )) and an endofunctor F : D fd ( A ) → D fd ( A ) , we have h t ( F ) = h σ,t ( F ) . ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 21 Next we describe h t ( F ) by using the Hillbert-Poincar´e polynomial. Definition 4.2. For a dg-module M ∈ D fd ( A ), define the Hilbert-Poincar´e poly-nomial of M by P t ( M ) := X k ∈ Z dim H k ( M ) e − kt ∈ Z [ e t , e − t ] . As in Section 3.5, we construct the special stability condition σ = ( Z , H s ) byusing the standard heart H s . Then by definition of σ , we have m σ ,t ( M ) = e t P t ( M )for any dg-module M ∈ D fd ( A ). As a result, the entropy is described as follows. Proposition 4.3. Let F : D fd ( A ) → D fd ( A ) be an endofunctor and G ∈ D fd ( A ) be a split-generator. Then the entropy of F is given by h t ( F ) = lim n →∞ n log P t ( F n G ) . Entropy of spherical twists. In this section, we compute the entropy ofSeidel-Thomas spherical twists on the derived categories of Calabi-Yau algebrasassociated with acyclic quivers. Let Q be an acyclic quiver with vertices { , . . . , n } and Γ N Q be the Ginzburg N -Calabi-Yau dg-algebra associated with Q for N ≥ N Q , see [Gin, Section 4.2] or [Kel11, Section 6.2].) Set D NQ := D fd (Γ N Q ). By [Kel11, Theorem 6.3], the category D NQ becomes a N -Calabi-Yau category , i.e. there is a natural isomorphismHom( E, F ) ∼ −→ Hom( F, E [ N ]) ∗ for E, F ∈ D NQ . (The notation V ∗ is the dual space of a K -linear space V .) Inthe Calabi-Yau category, we can consider a certain class of objects, called sphericalobjects. An object S ∈ D NQ is called N -spherical ifHom( S, S [ i ]) = ( K if i = 0 , N . For a spherical object S ∈ D NQ , Seidel-Thomas [ST01] defined an exact autoequiva-lence Φ S ∈ Aut( D NQ ), called a spherical twist , by the exact triangleHom • ( S, E ) ⊗ S −→ E −→ Φ S ( E )for any object E ∈ D NQ . The inverse functor Φ − S ∈ Aut( D NQ ) is given byΦ − S ( E ) −→ E −→ S ⊗ Hom • ( E, S ) ∗ . The Ginzburg dg-algebra Γ N Q satisfies the conditions in Section 4.1 when N ≥ N = 2, we need some modification.) Hence the category D NQ has the standard algebraic heart H s ⊂ D NQ generated by simple Γ N Q -modules S , . . . , S n corresponding to vertices { , . . . , n } of Q . In addition, these objects S , . . . , S n become N -spherical by [Kel11, Lemma 4.4]. Thus we can define spherical twistsΦ S , . . . , Φ S n ∈ Aut( D NQ ). In the following, we compute the entropy of sphericaltwists Φ S , . . . , Φ S n by using Proposition 4.3. For simplicity, write Φ i := Φ S i . Lemma 4.4. For a spherical twist Φ i ∈ Aut( D NQ ) and a spherical object S j ∈ D NQ , the Hilbert-Poincar´e polynomial of Φ ki S j ( k ≥ is given by P t (Φ ki S j ) = e k (1 − N ) t if i = j q ij P k − l =0 e l (1 − N ) t if q ij > 01 + q ji e (2 − N ) t P k − l =0 e l (1 − N ) t if q ji > otherwisewhere q ij is the number of arrows from i to j in Q . Proof. First we note thatdim Hom( S i , S j [ m ]) = i = j and m = 0 , Nq ij if q ij > m = 1 q ji if q ji > m = N − 10 otherwise . By the definition of spherical twists, it is easy to see that Φ ki S i = S i [ k (1 − N )] andhence P t (Φ ki S i ) = e k (1 − N ) t . If i = j and q ij = q ji = 0, then Φ ki S j = S j and hence P t (Φ ki S j ) = 1. Consider the case q ij > 0. SinceHom • ( S i , S j ) ⊗ S i = M m ∈ Z Hom( S i [ m ] , S j ) ⊗ S i [ m ] ∼ = S ⊕ q ij i [ − , we have an exact triangle S j → Φ i S j → S ⊕ q ij i → S j [1] . Applying the spherical twist Φ i for the above exact triangle repeatedly, we obtaina sequence of exact triangles S j / / Φ i S j (cid:1) (cid:1) ✄✄✄✄✄✄✄ / / Φ i S j { { ①①①①①①①① / / . . . / / Φ k − i S j / / Φ ki S j . x x qqqqqqqqqq S ⊕ q ij i [ [ ✼ ✼ ✼ S ⊕ q ij i [1 − N ] b b ❋ ❋ ❋ ❋ S ⊕ q ij i [( k − − N )] g g ◆ ◆ ◆ ◆ ◆ ◆ This implies the result in the case q ij > q ji > (cid:3) ASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY 23 Proposition 4.5. Let Q be a connected acyclic quiver and assume that Q is nota quiver with one vertex and no arrows. Then the entropy of spherical twists Φ , . . . , Φ n is given by h t (Φ i ) = ( if t ≥ − N ) t if t < . Proof. We use the generator G = ⊕ nj =1 S j . Then P t (Φ ki G ) = P nj =1 P t (Φ ki S j ).Recall from Proposition 4.4 that P t (Φ ki S j ) = 1 + q ij k − X l =0 e l (1 − N ) t = 1 + q ij − e k (1 − N ) t − e (1 − N ) t in the case q ij > P t (Φ ki S j ) = 1 + q ji e (2 − N ) t k − X l =0 e l (1 − N ) t = 1 + q ij e (2 − N ) t − e k (1 − N ) t − e (1 − N ) t in the case q ji > 0. First we consider the case t > 0. Then the above two termsconverge to some positive real numbers as k → ∞ since (1 − N ) t < 0. By theassumption of Q , the sum P nj =1 P t (Φ ki S j ) contains at least one of the above two.As a result, P nj =1 P t (Φ ki S j ) also converges to some positive real number as k → ∞ .Thus h t (Φ i ) = lim k →∞ k log P t (Φ ki G ) = 0when t > 0. Next consider the case t < 0. Similarly we can show that e − k (1 − N ) t P nj =1 P t (Φ ki S j )converges to some positive real number as k → ∞ since − (1 − N ) t < 0. Thus h t (Φ i ) = lim k →∞ k log P t (Φ ki G ) = lim k →∞ k log e k (1 − N ) t e − k (1 − N ) t P t (Φ ki G )= (1 − N ) t + lim k →∞ k log e − k (1 − N ) t P t (Φ ki G ) = (1 − N ) t when t < 0. Finally we can easily check that h t (Φ i ) = 0 when t = 0. (cid:3) Remark 4.6. The subgroup of autoequivalences generated by spherical twists Sph( D NQ ) := h Φ , . . . , Φ n i ⊂ Aut( D NQ ) is called the Seidel-Thomas braid group. Here we only computed the entropy ofgenerators Φ , . . . , Φ n . It is important problem to compute the entropy h t (Φ) for ageneral element Φ ∈ Sph( D NQ ) . Lower bound of the entropy on the derived categories of surfaces. Let X be a smooth projective variety over C and denote by D b ( X ) the boundedderived category of coherent sheaves on X . Define the Euler form χ : K (D b ( X )) × K (D b ( X )) → Z by χ ( E, F ) := X i ∈ Z ( − i dim C Hom D b ( X ) ( E, F [ i ]) . The numerical Grothendieck group N ( X ) is the quotient of K (D b ( X )) by the radicalof the Euler form χ . Let End F M (D b ( X )) be the semi-group consisting of Fourier-Mukai type endofunctors. Since these endofunctors preserve the radical of χ , theyinduce linear maps on N ( X ), i.e. the semi-group homomorphismEnd F M (D b ( X )) → End( N ( X )) , F [ F ]is well-defined (see [KT, Section 5.1]). A stability condition σ = ( Z, P ) is called numerical if Z : K (D b ( X )) → C factors through the numerical Grothendieck group N ( X ).In [Bri08, AB13], a numerical stability condition on D b ( X ) was constructed whendim C X = 2. Applying Theorem 3.5 and Proposition 3.11, we obtain the followinglower bound of the entropy. Proposition 4.7. Let X be a smooth projective surface over C and F : D b ( X ) → D b ( X ) be a Fourier-Mukai type endofunctor. 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