Categorical polynomial entropy

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 action on cohomology. We also establish in general a Yomdin-type lower bound for the categorical polynomial entropy of an endofunctor in terms of the induced endomorphism on the numerical Grothendieck group of the category. As examples, we compute the categorical polynomial entropy for some standard functors like shifts, Serre functors, tensoring line bundles, automorphisms, spherical twists, P-twists, and so on, illustrating clearly how categorical polynomial entropy refines the study of categorical entropy and enables us to study the phenomenon of categorical trichotomy. A parallel theory of polynomial mass growth rate is developed in the presence of Bridgeland stability conditions.

1. Introduction 1.1. Background. A topological dynamical system consists of a compact Hausdorff topological space X and a continuous self-map f : X → X. The topological entropy of f , denoted by h top (f ), is a non-negative real number (possibly infinite) measuring the complexity of the system, by looking at the asymptotic behaviour of the iterations of the map f . More precisely, using an auxiliary metric 1 on X, the topological entropy h top (f ), which is independent of the metric, is defined to be the exponential growth rate of certain positive number cov(f, n, ǫ) measuring the ǫ-separation property of the n-th iteration of f : We refer to [30,23,9] for more details. When restricting to the category of compact Kähler manifolds, the topological entropy can be understood via the cohomology; this is the content of the following fundamental result due to Gromov (the upper bound) and Yomdin (the lower bound).
Theorem 1.1 ( [25,48]). Let f be a C ∞ self-map of a compact differentiable manifold X. Then where ρ(f * ) denotes the spectral radius of the induced endomorphism f * on the cohomology H * (X, C). Moreover, if X is a compact Kähler manifold and f : X → X is a surjective holomorphic map, then h top (f ) = log ρ(f * ).
In the compact Kähler setting, a more refined study involves the so-called dynamical degrees (see the work of Dinh-Sibony [15,16]); some important facts on this matter are collected in Section 5.1. We refer to the survey of Oguiso [43] and the references therein for an overview of interesting examples and results on dynamical systems in algebraic geometry.
Recently, there has been a growing interest in dynamical systems with vanishing topological entropy, highlighting the study of the so-called polynomial entropy; see for instance [3,9,27,36,37,38,41,42]. More precisely, given a topological dynamical system (X, f ) with h top (f ) = 0, the polynomial entropy is defined to be the polynomial growth rate of the number cov(f, n, ǫ): h pol (f ) := lim ǫ→0 lim sup n→∞ log cov(f, n, ǫ) log(n) .
This provides a refined/secondary invariant for the dynamical system. In the spirit of Theorem 1.1, given an automorphism of a compact Kähler manifold with null entropy, Cantat-Paris-Romaskevich [9, Theorem 2.1 and Theorem 4.1] established some upper bounds for the polynomial entropy in terms of the induced action on cohomology, together with some topological data of the manifold. If certain dynamical degree is 1, the study of a secondary invariant called polynomial dynamical degrees is initiated by Lo Bianco [40,Section 1.3]. In Section 5.2 we extend his definition and results to the general case where the dynamical degree is arbitrary. We refer to [9] and references therein for other results on polynomial entropy in the compact Kähler setting.
The categorical counterpart of the notion of topological entropy was introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich [12]. The definition of categorical entropy is motivated by the profound connection between the Teichmüller theory and the theory of stability conditions on triangulated categories, established recently in [21,7,26,12]. Let D be a triangulated category and let F : D → D be a triangulated functor that we always assume not to be virtually zero, that is, F n ∼ = 0 for all n > 0. We view the pair (D, F ) as a categorical dynamical system. The categorical entropy of F , denoted by h t (F ), which is a real function in one real variable, is defined to be the exponential growth rate of certain positive number δ t (G, F n (G)) measuring the complexity of the image of a split generator G of D under the n-th iteration of F , with respect to G itself: Note that the categorical entropy is independent of the choice of the split generator. We review the definition and basic properties of δ t and h t in Section 2.1. The value of the categorical entropy at t = 0 is often of particular interest, and is denoted by The connection to the classical dynamical system justifies the notion of categorical entropy: it is proved in [12,Theorem 2.12] under some technical condition and by Kikuta and Takahashi [34] in full generality that if f : X → X is a surjective endomorphism of a smooth projective complex variety and Lf * : D b (X) → D b (X) is the derived pullback functor on the bounded derived category of coherent sheaves on X, then h cat (Lf * ) = h top (f ).
See Theorem 5.6 for a more general version due to Ouchi [45].
1.2. New invariants. The goal of the present article is to lay the foundation of the theory of categorical polynomial entropy, which should serve as the categorical counterpart of the aforementioned notion of polynomial entropy. Inspired by [9], we propose to define it as the polynomial growth rate of δ t (G, F n (G)): Note that h pol t (F ) is independent of the choice of the split generator G. Moreover, the definition makes sense even if the categorical entropy h t (F ) does not vanish, as long as h t (F ) = −∞. We refer to Section 2 for some basic properties of h pol t (F ). Its value at t = 0 is denoted by h pol (F ) := h pol 0 (F ) and is well-defined since h 0 (F ) ≥ 0 is a real number.
If the triangulated category D admits a Bridgeland stability condition σ, the notion of mass growth of an endofunctor F , denoted by h σ,t (F ), was defined in [12] and studied in [29], as a comparable invariant of categorical entropy. Mass growth is the categorical analogue of the volume growth for classical dynamical systems equipped with a Riemannian metric, see Yomdin [48]. We develop in Section 3 the basic theory of polynomial mass growth for a categorical dynamical system endowed with a stability condition (D, F, σ), in a parallel way to the theory of categorical polynomial entropy. The polynomial mass growth rate is invariant under deformation of the stability condition inside the stability manifold (Lemma 3.5).
Let us put those invariants into perspective, with the new ones colored in blue: Topological system (X,f : X→X)

Categorical system (D,F : D→D)
Entropy h pol (f ) h pol t (F ) Volume/mass growth lov(f ) h σ,t (F ) Polynomial volume/mass growth povol(f ) h pol σ,t (F ) 1.3. Gromov-Yomdin-type results. As a justification of our definition, when the categorical dynamical system comes from a classical one, namely, for a surjective endomorphism f of a smooth projective variety X defined over an algebraically closed field, it turns out that the categorical polynomial entropy behaves better than the (topological) polynomial entropy, in view of Theorem 1.1. More precisely, in Theorem 5.8, we show that the categorical polynomial entropy of Lf * can be computed using polynomial dynamical degrees (Definition 5.1), and is equal to the polynomial growth rate, in the generalized sense of Definition 4.1, of the induced action on the numerical Grothendieck group (or the cohomology in the complex setting) of X. Here in the introduction, let us only state the following special case over the complex numbers and with the additional assumption that the entropy vanishes.
Theorem 1.2 (see Corollaries 5.9 and 5.11). Let f be a surjective endomorphism of a smooth projective complex variety X. Assume that the topological entropy of f is zero. Then h pol t (Lf * ) is a constant function with value where f * ∈ End(H * (X, Z)) is the induced endomorphism on cohomology and s(f * ) + 1 is the maximal size of its Jordan blocks. If moreover D b (X) admits a numerical stability condition σ, then h pol σ,0 (Lf * ) is also equal to the above quantities.
One should compare the above theorem to the estimates on the (topological) polynomial entropy of automorphisms of compact Kähler manifold established by Cantat and Paris-Romaskevich [9, Theorem 2.1 and Theorem 4.1].
In general, we establish the following Yomdin-type lower bound for the categorical polynomial entropy. (i) If D is saturated, then h cat (F ) ≥ log ρ(N (F )) by [33,Theorem 2.13]. If the equality holds (for instance when h cat (F ) = 0), then where s(N (F )) is the polynomial growth rate of the linear map N (F ) (cf. Definition 4.1). (ii) If D admits a numerical stability condition σ, then h σ,0 (F ) ≥ log ρ(N (F )) by [29,Theorem 1.2].
If the equality holds (for instance when h σ,0 (F ) = 0), then h pol σ,0 (F ) ≥ s(N (F )). In Proposition 4.4 and Corollary 4.6, we show that the lower bounds are attained for any autoequivalence of the bounded derived category of a hereditary finite dimensional C-algebra.
1.4. Categorical trichotomy. The trichotomy for birational automorphisms of projective surfaces is one of the most fascinating phenomena in classical (algebraic) dynamical systems, see Cantat [8] and Diller-Favre [11]. Recall that given a smooth projective surface X, a birational self-map f : X X falls into three possibilities: elliptic, parabolic, and loxodromic. More precisely, let H be a polarization on X, which allows one to define the algebraic degree deg H (f ), then f is called • elliptic, if deg H (f n ) is bounded in n; in this case, f conjugates to a map that is virtually isotopic to the identity; • parabolic, if deg H (f n ) has polynomial growth in n; in this case, f preserves a fibration; • loxodromic, if deg H (f n ) has exponential growth in n; in this case, f has positive entropy and there are strong restrictions to the surface. We refer to [11] for more details.
By combining the force of categorical entropy and categorical polynomial entropy, we are naturally led to study the analogous categorical trichotomy by calling a categorical dynamical system (D, F ) We are indeed able to pinpoint some natural examples illustrating this phenomenon. The first instance we discover concerns the autoequivalences of derived categories of elliptic curves, refining Kikuta's result [32,Section 3.2]. As before, N (F ) is the induced endomorphism on the numerical Grothendieck group, which is 2-dimensional in the curve case.
Theorem 1.4 (see Theorem 6.21). Let D be the bounded derived category of a smooth projective curve of genus 1 defined over an algebraically closed field, and let F : D → D be an autoequivalence. Then A second instance of such categorical trichotomy is proved for the 3-Calabi-Yau category associated to the A 2 -quiver, by looking at the group of autoequivalences generated by the two natural spherical twists, see Section 6.4 for the precise statement.  [46], and P-twists along P-objects (analogue of Dehn twists along Lagrangian complex projective space) [28]. We show in Section 6 that the categorical polynomial entropy provides a non-trivial invariant for these autoequivalences. For example, the categorical polynomial entropy of the autoequivalence of tensoring a line bundle in the derived category of a smooth projective variety encodes the positivity property of the line bundle. Theorem 1.5 (see Proposition 6.4, Theorem 6.7 and Corollary 6.9). Let L be a line bundle on a smooth projective variety X defined over an algebraically closed field. The categorical polynomial entropy of the functor − ⊗ L on D b (X) is a constant function whose value depends only on the numerical class of L, and satisfies We refer to Sections 6.3, 6.4, and 6.5 for results on the categorical polynomial entropy of spherical twists and P-twists.
Finally, we observe the following Gromov-Yomdin-type equality in the curve case, which is reminiscent of [32]. Theorem 1.6 (See Proposition 6.19 and Theorem 6.21). Let C be a smooth projective curve defined over an algebraically closed field. Let F be any autoequivalence of the bounded derived category D b (C). Then the categorical polynomial entropy of F is equal to the polynomial growth of the induced action on the (2-dimensional) cohomology/numerical Grothendieck group: Convention: k is a base field. In this paper, all categories, as well as functors between them, are assumed to be k-linear. Functors between triangulated categories are always assumed to be triangulated (i.e. they preserve distinguished triangles) and not virtually zero (i.e. no iteration is the zero functor).
A triangulated category is called saturated if it is equivalent to the homotopy category of a triangulated saturated A ∞ -category; or equivalently, it admits a dg-enhancement which is triangulated, smooth and proper. Note that the notion of saturatedness here is stronger than that of Bondal-Kapranov [4].

Acknowledgement:
We would like to thank Serge Cantat, Zhi Jiang, John Lesieutre, Olga Paris-Romaskevich, Steffen Sagave, and Junyi Xie for helpful discussions. Lie Fu and Genki Ouchi also want to thank the fourth edition of Japanese-European Symposium on Symplectic Varieties and Moduli Spaces in Zürich 2019, where the initial ideas of this project took form. Genki Ouchi is supported by Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) in RIKEN and JSPS KAKENHI Grant number 19K14520.

Categorical polynomial entropy
In this section, inspired by the categorical formalism from [12] and geometrical considerations from [9], we develop the basic theory of categorical polynomial entropy of an endofunctor of a triangulated category, as a secondary invariant of a categorical dynamical system, refining the categorical entropy. • the complexity function of N with respect to M : (i) When D is saturated, the complexity function and the Ext-distance function are controlled by each other: there exist functions C 1 , C 2 : R → R >0 , such that for any M, N ∈ D, we have (ii) They both satisfy the subadditivity condition: for any distinguished triangle (iii) δ t satisfies the (multiplicative) triangle inequality: (iv) δ t retracts after applying a functor: for any triangulated functor F : D → D ′ , we have (v) If D admits a Serre functor S, then by Serre duality, Using the complexity functions, Dimitrov-Haiden-Katzarkov-Kontsevich [12] defined a categorical analogue of the notion of topological entropy, in order to measure the complexity of an endofunctor, viewed as a categorical dynamical system. Let us recall the definition. n .
The existence of the limit is proved in [12, Lemma 2.6]. As the notation suggests, h t (F ) is independent of the choice of the generator [12, Lemma 2.6]. We denote Inspired by [9], even in the case of "slow" categorical dynamical systems, namely, when the categorical entropy vanishes, it is interesting to understand its complexity. To this end, we propose the following notion as a secondary invariant, which makes sense even when the categorical entropy does not vanish. Definition 2.4 (Categorical polynomial entropy). In the same setting as in Definition 2.3, we define the polynomial entropy of F to be the function h pol t (F ) in the real variable t given by where h t (F ) is the categorical entropy of F . It is well-defined for any t ∈ R such that h t (F ) = −∞.
Remark 2.5 (Lower polynomial entropy). It also makes sense to use lim inf instead of lim sup in Definition 2.4. The function thus obtained could be called the lower polynomial entropy of F , denoted by h pol t (F ). We do not know whether h pol t (F ) and h pol t (F ) coincide in general. In this paper, we will mostly focus on the study of h pol t .
2 An object of a triangulated category is called a split generator if the smallest thick triangulated subcategory containing it is the whole category.

Basic properties.
We show some basic properties of the categorical polynomial entropy function, in a parallel way to the analogous properties of the categorical entropy, as developed in [12], [32], [34].
Lemma 2.6. The definition of the categorical polynomial entropy is independent of the choice of the split generator. Moreover, for any two split generators G, G ′ of D, we have Proof. The proof is similar to [12, Lemma 2.6] by using the triangle inequality and the retraction property of δ t , recalled in Remark 2.2 (iii), (iv).
With the saturatedness condition on the category, one can use the Ext-distance function to compute the categorical polynomial entropy.
Lemma 2.7. Assume that D is saturated. For any split generators G, G ′ of D, we have that Proof. As in [12,Theorem 2.7], one uses Remark 2.2 (i) and Lemma 2.6.
Lemma 2.8. Notation is as before. Assume that D is saturated and there is a split generator G of D and an integer M ≥ 0 such that for any |k| ≥ M and any n ≥ 0, we have Ext k (G, F n (G)) = 0, (for example, when F preserves a bounded t-structure of finite cohomological dimension), then h pol t (F ) is a constant function in t.
We can conclude by dividing by log(n) and taking limit. Lemma 2.9 (Commutation and conjugation). Let D be a triangulated category and F 1 , F 2 two endofunctors of D. Then h pol . Proof. As shown in [32,Lemma 2.8], h t (F 1 F 2 ) = h t (F 2 F 1 ) and for any split generator G, Hence . We get an equality by symmetry. The invariance by conjugation follows: Lemma 2.10 (Powers). Notation is as before. For any positive integer m, we have In particular, if the lim sup in the definition of h pol t (F ) is an actual limit, then we have equalities.
Proof. By definition, Using the fact that h t (F m ) = mh t (F ), the right-hand side is nothing else but which is less than or equal to h pol t (F ). The proof for the lower polynomial entropy is similar.
Lemma 2.11 (Inverse). Let D be a saturated triangulated category admitting a Serre functor. Then for any autoequivalence F of D, we have Proof. Let G be a split generator and S the Serre functor. Then for any n > 0, where the second equality follows from Remark 2.2 (v) and the fact that all autoequivalences commute with the Serre functor. As S(G) is also a split generator of D, we obtain that Consequently, combined with Lemma 2.7, The following observation will be used in Section 6.2 when we study the categorical polynomial entropy of tensoring a line bundle. Lemma 2.12 (Composition of commuting functors). Let D be a triangulated category. Let F 1 be an autoequivalence and F 2 be an endofunctor. Assume that In this case, In this case, Proof. We only show (i), as the proof of (ii) is similar. By Remark 2.2 (iii) and (iv), for any split generator G, we have , where the last equality follows from Lemma 2.11. Hence When h t (F 1 ) is an odd function, we get the claimed equality. In this case, Then for any non-zero object E, its mass function with respect to σ is defined as the following real function in t: where A k are the σ-semistable factors of E and φ is the phase function. Denote m σ := m σ,0 . The space of stability conditions is denoted by Stab(D) (or more precisely Stab Γ (D) if one want to specify the choice of Γ), which is naturally a complex manifold of dimension rk(Γ), by [5].
Remark 3.1. We collect some fundamental properties of the mass function, due to [12] and [29].
where δ t is the complexity function in Definition 2.1. (iii) If the distance (defined in [5]) between two stability conditions σ, τ is finite, then there exist two functions C 1 (t), C 2 (t) : R → R >0 , such that for any non-zero object E, we have Recall also the definition of the mass growth in [12, Section 4.5].
Definition 3.2 (Mass growth). Let D be a triangulated category endowed with a stability condition σ. Let G be a split generator of D, then the mass growth function of an endofunctor F is defined as It is shown in [29,Theorem 3.5] that the definition is independent of the choice of the split generator, and Moreover, it depends only on the connected component of Stab(D) in which σ lies [29, Proposition 3.10]. We denote h σ (F ) := h σ,0 (F ), the value at t = 0.
We propose the following notion as a secondary measurement of the mass growth, similarly to Definition 2.4. Definition 3.3 (Polynomial mass growth). In the same setting as in Definition 3.2, the polynomial mass growth function of F is defined to be Proof. The proof is the same as in [29, Theorem 3.5(1)], by using Remark 3.1 (ii). Proof. Similarly as in [29,Proposition 3.10], it is a direct consequence of Remark 3.1 (iii).
Given an endofunctor, to relate its mass growth to its entropy, Ikeda [29, Theorem 3.5 (2)] showed that h σ,t (F ) ≤ h t (F ). The following is its polynomial counterpart.
One can conclude by dividing by log(n) and taking limit.
Recall that a stability condition σ = (Z, A) is called algebraic if the corresponding heart A is a finite length abelian category with finitely many isomorphism classes of simple objects. Examples of triangulated categories admitting algebraic stability conditions include derived categories with full strong exceptional collection, derived categories of (homologically) finite-dimensional dg-modules over a connective dg-algebra of finite type etc.
The proof is similar to [29, Theorem 3.14]. Ikeda proved h σ,t (F ) = h t (F ) by showing more strongly that there exists a special algebraic stability condition σ 0 in the same connected component of the stability manifold, such that which allows us to deduce that h pol σ0,t (F ) = h pol t (F ). One concludes by Lemma 3.5.

Yomdin-type Estimates
In the spirit of Gromov-Yomdin's Theorem 1.1, given an endofunctor of a triangulated category, we want to understand its polynomial entropy and its polynomial mass growth rate, which are of categorical nature, in terms of some cohomological data, which is essentially a matter of linear algebra. , where s(M ) + 1 is the size of maximal Jordan blocks with eigenvalues having maximal modulus; we feel that it is meaningful to call s(M ) the polynomial growth rate of M . To make this idea precise, we propose the following definition, which generalizes the notion of polynomial growth rate used in [9, Section 2], by normalizing the exponential growth rate determined by the spectral radius.
Definition 4.1 (Polynomial growth rate). Let φ be an endomorphism of a finite-dimensional vector space endowed with some norm − . The polynomial growth rate of φ is defined to be As all norms on the space of matrices are equivalent, s(φ) is independent of the choice of the norm.
Let us record the following basic result.
Notation is as before. The limit in Definition 4.1 exists, and it is precisely one less than the maximal size of the Jordan blocks whose eigenvalues are of maximal absolute value ρ(φ). In paricular, s(φ) is a natural number.
Proof. Let φ ρ(φ) = D + N be the Jordan decomposition, where D is semisimple, N is nilpotent and N D = DN . Then the eigenvalues of D are of modulus ≤ 1. Let s + 1 be the maximal size of the Jordan blocks of φ ρ(φ) whose eigenvalues are of maximal modulus 1. We have where, on the right-hand side, the norm of the first term has growth at most O(n s−1 ), the norm of the second term has growth equivalent to n s , and the third term tends to zero, when n → ∞. Therefore, φ n ρ(φ) n has growth equivalent to n s .

4.2.
Lower bound for categorical polynomial entropy. Given a saturated triangulated category D, its numerical Grothendieck group, denoted N (D), is by definition the quotient of the Grothendieck group K 0 (D) by the radical of the Euler pairing χ(E, We establish the following Yomdin-type lower bound for the categorical polynomial entropy in terms of the induced action on the numerical Grothendieck group. In passing, we provide an alternative proof for the lower-bound of the categorical entropy previously obtained by [33,Theorem 2.13]. If the equality holds (for example when h cat (F ) = 0), then where s is the polynomial growth rate.
Proof. For ease of notation, denote f := N (F ) ∈ End(N (D)). Let λ be an eigenvalue of f with |λ| = ρ(f ) such that its characteristic space ker(f − λ id) ∞ has a maximal Jordan block, whose size is Take objects M 1 , . . . , M m in D such that their classes in N (D) form a basis. We define the following norm on N (D) C : In general, the inequality in Proposition 4.3 can be strict, see Example 6.8. We give here an example where the previously established lower bound is achieved. More examples will be presented in Section 6.
Recall that an associative algebra is called hereditary if its has global dimension at most 1. Important examples of hereditary algebras include semisimple algebras, path algebras of finite quivers without oriented cycles etc. Proof. There are projective A-modules P 1 , · · ·, P d such that P 1 , · · ·, P d is a full strong exceptional  More generally, if the central charge of the stability condition σ factors through cl : K 0 (D) → Γ for some lattice Γ, and suppose that the homomorphism cl is surjective and its kernel is preserved by F . Then Proof. We only prove the case where the central charge Z factors through N (F ), the general case is similar. We proceed as in the proof of Proposition 4.3. Keeping the same notation there and using the fact that m σ (−) ≥ |Z(−)|, we have that for any n > 0, If h σ (F ) = log |λ|, assume moreover that Z(v s ) = 0, since G ⊕ E is a split generator, we have the following, by Lemma 2.7, The condition Z(v s ) = 0 can always be achieved by deforming the stability condition σ (note that dim Stab(D) = dim N (F )), and this does not affect h σ (F ) or h pol σ (F ), thanks to Lemma 3.5.
Therefore, Lemma 3.6 and Proposition 4.5 imply that s(N (F )). Then Proposition 4.4 allows us to conclude the proof.

Classical dynamical systems: a categorical retake
In this section, we make a connection of the categorical theory developed so far to the classical setting. Let k be an algebraically closed field and let X be a smooth projective k-variety endowed with a surjective (regular) endomorphism f : X → X. Note that f is automatically finite and flat. There have been extensive study of the complexity of the system (X, f ) by topological, geometric, algebraic, analytic and even probabilistic approaches. We employ here a categorical method (as in [12], [34]) by looking at the naturally associated categorical dynamical system (D b (X), Lf * ).

Dynamical degrees.
We recall some basic properties of a series of fundamental invariants, called dynamical degrees, of an algebraic/complex dynamical system. The material here is well-established in the literaure. We are in the following more broad setting: • f is a dominant rational self-map of a normal projective variety X defined over k, or • when k = C, f is a dominant meromorphism self-map of a compact Kähler manifold X. Let f : X X be as above. Denote by d the dimension of X. For any integer 0 ≤ p ≤ d, the p-th dynamical degree of f , denoted by d p (f ), is by definition where ω is the first Chern class of an ample line bundle (or a Kähler class in the compact Kähler setting). The definition is independent of the choice of ω and it is invariant under birational conjugation. The existence of the limit is due to Dinh-Sibony [16,14] when k = C, and to Truong [47] (see Dang [10] for an alternative treatment) when char(k) is arbitrary. Let N p := N p (X) be the group of algebraic cycles of codimension p on X modulo numerical equivalence. For a rational self-map g of X, we denote by g * N p the induced endomorphism on N p (X). Thanks to [10, Theorem 2], we have which is equal to ρ(f * N p ) when f is regular, where ρ denotes the spectral radius. Similarly, in the compact Kähler setting, we have which is equal to ρ(f * H p,p ) when f is holomorphic (or more generally, algebraically stable), where H p,p is the (p, p)-part in the Hodge decomposition of H 2p (X, C).
By the Teisser-Khovanskii inequality (see [24]), the sequence {d p (f )} d p=0 is log-concave ( [14,47,10]). When k = C, a celebrated theorem of Gromov-Yomdin [25,48] (cf. Theorem 1.1) says that when f is holomorphic (and surjective), the topological entropy can be computed from the dynamical degrees as well as the action on cohomology: As a consequence, f has positive topological entropy if and only if d 1 (f ) > 1.

Polynomial dynamical degrees.
Keep the same setting as in Section 5.1. Consider f : X X as before. In the Kähler situation, under the hypothesis that the topological entropy of f is zero, Lo Bianco [40, Section 1.3] initiated the study of the so-called polynomial dynamical degrees, the polynomial counterpart of dynamical degrees discussed in Section 5.1, for holomorphic endomorphisms, which was later extended by Cantat-Paris-Romaskevich [9, Section 3] to meromorphic self-maps.
Using the idea of normalizing the exponential growth as in Definition 4.1, we generalize this notion of polynomial dynamical degrees to rational/meromorphic self-maps of arbitrary entropy over arbitrary algebraically closed base field k: Definition 5.1 (Polynomial dynamical degree). Let f : X X be as above. For any integer 0 ≤ p ≤ d, the p-th polynomial dynamical degree, denoted by s p (f ), is by definition where ω is the first Chern class of an ample line bundle (or a Kähler class when k = C) and d p (f ) is the p-th dynamical degree of f . We will see shortly in Proposition 5.3 that if f is regular/holomorphic, then the lim sup is actually a limit, and takes values in N, the set of non-negative integers. We do not know whether the lim sup is a limit in general.
Recall the notion of polynomial growth rate in linear algebra in Definition 4.1.
Proposition 5.3. Notation is as above. If f is moreover regular, then for any 0 ≤ p ≤ d, the lim sup in Definition 5.1 is a limit, and Similarly, in the compact Kähler setting, if f is moreover holomorphic, then Proof. By [10, is convergent to a natural number by Lemma 4.2, the sequence in Definition 5.1 is also convergent with the same limit.
In the sequel when discussing the concavity properties, we have to stay in the following more restrictive setting: • f is a surjective (regular) endomorphism of a smooth projective variety X defined over k, or • f is a surjective holomorphic endomorphism of a compact Kähler manifold X.
Lemma 5.4 (Concavity). Let f : X → X be as above. The sequence {s p (f )} p1 p=p0 is concave. Proof. Let λ = d p0 (f ) = · · · = d p1 (f ). By the Teisser-Khovanskii inequality ( [24]), for any n > 0, is a concave sequence for p 0 ≤ p ≤ p 1 . One concludes by dividing by log(n) and taking limit. Note that the argument does not apply to the more general case where f is only assumed to be rational/meromorphic, as lim sup does not preserve the concavity.
Proposition 5.5. Notation is as before (in particular, f is regular). Let N * := p N p (X). We have Similarly, in the compact Kähler setting, Proof. Let λ = max p d p (f ). The case for N * is almost immediate: s p (f ).
In the compact Kähler situation, we need to show in addition that for any 0 , then it is easy to see (cf. loc. cit.) that there exists a constant C > 0 such that for any n > 0, Let us show that this is actually an equality and compute the corresponding polynomial dynamical degree of (f, f ). For any Kähler form ω on X, π * 1 ω + π * 2 ω is a Kähler form on X × X, where π 1 and π 2 are the natural projections from X × X to X.
where the l-th term has growth in n equivalent to d l (f ) n n s l (f ) d i+j−l (f ) n n s i+j−l (f ) , hence the sum has growth equivalent to λ 2n n s , with By the concavity of the sequence s p0 , . . . , s p1 (Lemma 5.4), if i+j = 2p is even, then s i+j (f, f ) = 2s p (f ); if i + j = 2p + 1 is odd, then s i+j (f, f ) = s p (f ) + s p+1 (f ). In any case, s i+j (f ) ≤ 2 max p0≤p≤p1 s p (f ). Now use again (5.1), we obtain that s p (f ).

5.3.
Using derived categories. Let f : X → X be a surjective endomorphism of a smooth projective variety X defined over k. Let D b (X) be the bounded derived category of coherent sheaves on X. Let Lf * : D b (X) → D b (X) be the derived pullback functor. As f is flat, Lf * = f * , but we will continue writing Lf * in the sequel to remind that it is a functor and to avoid confusion with the induced action on cohomology. When k = C, Kikuta-Takahashi [34] showed that its categorical entropy coincides with the topological entropy of f : This result was later extended over any algebraically closed field by Ouchi [45,Theorem 5.2]. Let us state their results as follows, incorporating also the discussion in Section 5.1 as well as some contribution from Ikeda [29].
Theorem 5.6 (Kikuta-Takahashi, Ouchi). Let f : X → X be a surjective endomorphism of a smooth projective variety X defined over an algebraically closed field k. Then h t (Lf * ) is constant with value If k = C, they are also equal to log ρ(f * H * ) = h top (f ). Moreover, if D b (X) admits a stability condition, then the previous quantities are equal to the mass growth rate h σ (Lf * ) for any numerical stability condition σ. As a consequence, ρ(N (Lf * )) = ρ(f * N * ) and s(N (Lf * )) = s(f * N * ).
The main result of this section is the following analogy of Theorem 5.6 for the polynomial entropy.
Theorem 5.8. Let f be a surjective endomorphism of a smooth projective variety X defined over an algebraically closed field k. Let Lf * : D b (X) → D b (X) be the derived pullback functor. Then the categorical polynomial entropy function of Lf * is constant with value , where the maximum runs over all integers p such that d p (f ) attains the maximal dynamical degree of f . Here s p (f ) denotes the p-th polynomial dynamical degree of f (see Section 5.2), and s(f * N * ) is the polynomial growth rate of the induced action on the numerical Chow group N * (X) := p N p (X).
Proof. First of all, the fact that the function h pol t (Lf * ) is constant follows from Lemma 2.8, since Lf * = f * preserves the standard t-structure on D b (X). Let d be the dimension of X. By Fujita's vanishing theorem ( [20], see also [39, 1.4.35]), there exists a very ample line bundle L such that for all i > 0 and all nef line bundle L ′ . Take split generators G = d+1 j=1 L −j and G ′ = G ∨ = d+1 j=1 L j of D b (X) ( [44,Theorem 4]). By the choice of L, we have that for all n > 0 and all i > 0.
Therefore, we obtain that Since χ(−, −) is a non-degenerate bilinear form on N (X), there exists a constant C > 0 such that

It yields that
where the first equality uses Lemma 2.7 (note that D b (X) is saturated), and the second equality uses Theorem 5.6 (and Remark 5.7).
On the other hand, by Theorem 5.6, the hypothesis in Proposition 4.3 is satisfied. Hence we get the Yomdin-type lower bound: We conclude that this is an equality; hence, h pol (Lf * ) = s(f * N * ) by Remark 5.7. The remaining assertions follow from Proposition 5.5.
Combining the previous theorem with our discussion on polynomial dynamical degrees in Section 5.2, we deduce the following consequence. Corollary 5.9 (Complex setting). If f is a surjective holomorphic endomorphism of a projective complex manifold X, then the categorical polynomial entropy of Lf * is equal to the polynomial growth rate of the induced action f * on the cohomology H * (X, Q): In particular, if the topological entropy of f is zero, then h pol t (Lf * ) is a constant function with value where s(f * ) + 1 is the maximal size of the Jordan blocks of f * .
Proof. The first part can be easily deduced by combining Proposition 5.5 with Theorem 5.8. Now assuming the vanishing of the topological entropy, the Gromov-Yomdin Theorem 1.1 implies that all dynamical degrees are equal to 1 and the spectral radius of f * on H * (X) is 1. Thanks to the integral structure on cohomology, all the eigenvalues of f * are algebraic integers, hence they are roots of unity by Kronecker's theorem. Therefore, f * is virtually unipotent and h pol (Lf * ) = s(f * ) has the description by Jordan blocks in the statement, thanks to Lemma 4.2.

Examples
We compute in this section the categorical polynomial entropy, as well as the polynomial mass growth rate of some standard functors, in a parallel way to [12,Section 2]. Recall that when concentrating in the value of entropy functions at t = 0, we write Shifts. The following lemma shows that cohomological shifts do not affect the polynomial entropy.
Therefore, using Lemma 2.7, we see that Similarly for the mass growth, we have the following result.  As for the polynomial entropy, we claim that h pol t (S) = 0. Indeed, fixing a split generator G of D, for any t ∈ R and any N > 0, if one writes N = nq + r with q = ⌊ N n ⌋ and 0 ≤ r < n, then we have ǫ t (G, S N (G)) = ǫ t (G, S r (G)[mq]) = ǫ t (G, S r (G)) · e mqt . Therefore log ǫ t (G, S N (G)) − N · h t (S) = log ǫ t (G, S r (G)) − mr n t.
To conclude, it suffices to observe that the absolute value of the right-hand side is bounded, independently of N , by max Similarly, for the mass growth rate, instead of assuming the saturatedness, we suppose there exists a stability condition σ on D, then h σ,t (S) = m n t and h pol σ,t (S) = 0.
As a consequence, for any endofunctor F commuting with S (for example, an autoequivalence), Lemma 2.12 implies that 6.2. Tensoring line bundles. Let the base field k be algebraically closed. Let X be a smooth projective variety defined over k and L a line bundle on X. We consider here the autoequivalence of tensoring with L on the derived category of X: It is shown in [12] that the categorical entropy function of this functor is the constant zero function: We shall study its categorical polynomial entropy, which is indeed a non-trivial and meaningful invariant. The first result below estimates the polynomial entropy. Recall that the numerical dimension of L is by definition where ≡ denotes the numerical equivalence relation.
Proposition 6.4. Notation is as before. Let d = dim(X).
(i) The polynomial entropy function h pol Proof. (i) follows from Lemma 2.8, since − ⊗ L preserves the standard t-structure of D b (X). (ii). By definition and (6.1), the value of this constant function is where G is any split generator. In the sequel, we choose G to be a locally free sheaf on X, for example is an isomorphism of finite-dimensional Q-vector spaces, making the following diagram commutative: where the bottom arrow is the intersection product with It is then clear that the spectral radius of ·e c1(L) ∈ End(N * (X)) is 1 and its polynomial growth rate is ν(L). The claim is proved. Combined with the Yomdin-type lower bound established in Proposition 4.3 (note that h cat (− ⊗ L) = 0), we obtain that h pol (− ⊗ L) ≥ s(·[L]) = ν(L).
Now we try to determine precisely the value of the polynomial entropy of the functor − ⊗ L. It turns out to be quite related to the positivity properties of L. For illustration, let us first show the following result on big and nef line bundles. Lemma 6.5 (Nef and big line bundles). Let L be a nef line bundle on a smooth projective variety X.
Proof. This is a special case of Theorem 6.7 below, but we prefer to give a direct proof here. If L is big and nef, then by [39,Corollary 1.4.41], dim H 0 (X, G ∨ ⊗ G ⊗ L ⊗n ) grows exactly as a polynomial of degree d. Hence ǫ(G, G ⊗ L ⊗n ) grows at least as a polynomial of degree d. Therefore h pol (− ⊗ L) ≥ d. We can conclude as Proposition 6.4 (iii) provides the other inequality. If L is nef but not big, on one hand, [39,Theorem 1.4.40] implies that the growth of the k-th term of (6.2) is at most like a polynomial of degree d − k: On the other hand, the hypothesis implies that (c 1 (L) d ) = 0, then [39,Corollary 1.4.41] says that the 0-th term of (6.2) also grows at most as a polynomial of degree d − 1. As a result, ǫ(G, G ⊗ L ⊗n ) grows at most as a polynomial of degree d − 1. Therefore h pol (− ⊗ L) ≤ d − 1.
To generalize Lemma 6.5, we first establish the following result on the growth of cohomology of powers of nef line bundles, which will be used in the proof of Theorem 6.7. It is probably well-known by experts, but as we cannot find a reference, a proof is provided for the sake of completeness. Proposition 6.6. Let X be a smooth projective variety over an algebraically closed field k. Let L be a nef line bundle and F be a coherent sheaf on X. Then dim H k (X, F ⊗ L ⊗n ) = O(n ν(L) ), that is, there exists a constant C > 0 such that for all k ≥ 0 and n ≥ 0, we have where ν(L) is the numerical dimension of L.
Proof. The proof is similar to [39,Theorem 1.4.40]. We proceed by induction on the dimension of X. By Fujita's vanishing theorem [20], there is a very ample divisor H such that for all k > 0 and all n ≥ 0. Up to replacing H by a general member in its linear system, we can assume that H is smooth and does not contain any subvariety of X defined by associated primes of F . Therefore we have the exact sequence for any n ≥ 0. Let us look at the corresponding long exact sequence of cohomology groups. For any k > 0, by the vanishing hypothesis (6.3), we have which is O(n ν(L|H ) ) by the induction hypothesis. As ν(L| H ) ≤ ν(L), we conclude that dim H k (X, F ⊗ L ⊗n ) = O(n ν(L) ). For k = 0, again by (6.3), Thanks to the Hirzebruch-Riemann-Roch formula, where α i is some i-dimensional algebraic cycle of X. As c 1 (L) i ≡ 0 for all i > ν(L), the above polynomial is of degree at most ν(L). Therefore we can conclude that dim H 0 (X, F ⊗L ⊗n ) = O(n ν(L) ). The induction process is complete.
We are ready to determine the polynomial entropy for tensoring a nef line bundle.
Theorem 6.7 (Nef line bundles). Let L be a line bundle on a smooth projective variety X defined over an algebraically closed field k. If L or L −1 is nef, then the polynomial entropy of the functor − ⊗ L is equal to the numerical dimension of L: Proof. We only need to treat the case where L is nef, as the anti-nef case follows from the nef case by using Lemma 2.11. Keep the notation in the proof of Proposition 6.4. Applying Proposition 6.6 to the case F = G ∨ ⊗ G, we obtain that each term on the right-hand side of (6.2) is O(n ν(L) ), thus so is ǫ(G, G ⊗ L ⊗n ). Consequently, h pol (− ⊗ L) ≤ ν(L). Combined with the lower bound in Proposition 6.4 (iii), we must have an equality.
The nefness assumption in Theorem 6.7 can not be removed. Let us give a simple example of line bundle whose polynomial entropy is strictly bigger than its numerical dimension. Example 6.8. Let S be a smooth projective surface and H be an ample divisor on it. Assume that the degree of S is a square, that is, (H 2 ) = m 2 for some m ∈ Z >0 ; for example when S = P 2 and H = c 1 (O P 2 (1)). Let τ : S ′ → S be the blow up of S at a point. Let H ′ := τ * (H) be the pullback of the polarization and E the exceptional divisor. We have that (H ′ 2 ) = m 2 , (H ′ · E) = 0, and (E 2 ) = −1.  [44]). Then ǫ( , which has quadratic growth rate in n, since D is a big divisor. Hence h pol (− ⊗ O S ′ (D)) = 2. By taking product of S ′ with another variety and pulling back the divisor to the product, one can also produce examples where the polynomial entropy is strictly between the numerical dimension and the dimension. Remark 6.11 (Serre functor for a variety). Let X be a smooth projective variety. The Serre functor of the derived category D b (X) is given by S X = − ⊗ ω X [dim(X)]. Its categorical entropy is computed in [12, §2.6.2], namely, h t (S X ) = dim(X)t. As for its polynomial entropy, we see a close link with the birational geometry of X: using Lemma 6.1, Proposition 6.4, and Theorem 6.7, we obtain that which is • between the numerical dimension of ω X and dim(X); • equal to dim(X) if X is minimal and of general type; • equal to dim(X) if ω ∨ X is nef and big, i.e. X is a weak Fano variety; • equal to the numerical dimension of ω X if X is a minimal model. Assuming the abundance conjecture when char(k) = 0, then it is equal to the Kodaira dimension of X. The most interesting feature of spherical objects is that they induce autoequivalences of the triangulated category [46,2], called spherical twists. More precisely, for a d-spherical object E, the spherical twist around E is the autoequivalence

It follows from definition that
To study the polynomial entropy of spherical twists, we need the following estimate, which improves upon [45, Proof of Theorem 3.1]. Lemma 6.12 (Upper bound). For any objects G, G ′ ∈ D and any positive integer n, we have where A t = ǫ t (G ′ , RHom(E, G) ⊗ E) and B t = ǫ t (G ′ , G) are positively valued functions, which are independent of n.
Proof. For any n > 0, applying T n−1 E to the distinguished triangle The cases when d = 1 or t = 0 follow immediately. Assume now d ≥ 2 and t = 0. Then One can conclude easily by separating the cases t > 0 and t < 0. Proof. In any of the cases (i) ∼ (iv), Ouchi [45] computed the categorical entropy of the spherical twist T E : Fix two split generators G, G ′ of D. We first establish, in any case of (i) ∼ (iv), the lower bound that As G ⊕ E is also a split generator, Here we used the fact that As for (iv), take E a non-zero object in E ⊥ , as G ⊕ E is also a split generator, Let us now establish the upper bounds in the statement: (i) When d = 1, h t (T E ) = 0 and Lemma 6.12 says that Therefore, (ii). When t = 0, Lemma 6.12 implie that (iii). When t < 0 and d ≥ 2, using Lemma 6.12, we see that (iv) When t > 0 and d ≥ 2, assuming E ⊥ = 0, then Lemma 6.12 yields that Remark 6.14. When D is the derived category of a projective K3 surface, the condition on the orthogonal complement in Proposition 6.13 (iv) is often satisfied (for example when Picard number is 1), see Bayer's appendix of [45].
6.4. Spherical twists (II): quiver Calabi-Yau categories. Let Q be an acyclic quiver with vertices labelled by {1, 2, . . . , n}, and let D Q be the 3-Calabi-Yau category constructed from the Ginzburg Calabi-Yau dg-algebra associated to the quiver Q [22,31]. For each vertex 1 ≤ i ≤ n, there is an associated spherical object S i ∈ D Q . We denote T i the spherical twist associated to S i . The morphisms between spherical objects are determined by the quiver: if there are e ij arrows from vertex i to vertex j, then Hom • (S i , S j ) = C ⊕eij [−1]. By Lemma 3.7 and the same computations as in [29, Section 4.2], we have for any power k of the spherical twist T i . Here ℓ(n) is a linear polynomial which depends on k and the valency of the i-th vertex.
We focus on the case of the 3-Calabi-Yau category D = D A2 associated to the A 2 -quiver The subgroup of Aut(D) generated by the spherical twists T 1 , T 2 is isomorphic to the standard braid group on 3 strings T 1 , T 2 ∼ = Br 3 = σ 1 , σ 2 : σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 , cf. [6, Section 2]. There is a short exact sequence where the map Z → T 1 , T 2 is given by sending 1 to (T 1 T 2 ) 3 = [5], and the map is given by the induced action on the numerical Grothendieck group N (D): We prove the following trichotomy. (iii) h cat (F ) > 0 if and only if N (F ) is hyperbolic (i.e. |tr(N (F ))| > 2). In this case, F is pseudo-Anosov in the sense of [19]. Moreover, the Gromov-Yomdin-type equality holds for both categorical entropy and categorical polynomial entropy of any F ∈ T 1 , T 2 : h cat (F ) = log ρ(N (F )) and h pol (F ) = s(N (F )).
Proof. (i). An elliptic element in PSL(2, Z) is conjugate to either Observe that Since are of finite order up to shifts, hence F n = [m] for some integers n, m. In this case, we have for some n ∈ Z. Hence if N (F ) is parabolic and N (F ) = ±id, then F = gT n 1 g −1 [k] for some g ∈ T 1 , T 2 , n ∈ Z\{0}, and k ∈ Z. By the previous computations, we have h cat (F ) = 0 and h pol (F ) = 1.
Since N (F ) in this case is quasi-unipotent and has a single Jordan block of size 2, we have h cat (F ) = log ρ(N (F )) = 0 and h pol (F ) = s(N (F )) = 1.
(iii). The spectral radius of a hyperbolic element is greater than 1. Thus, if N (F ) is hyperbolic, then h cat (F ) ≥ log ρ(N (F )) > 0. We will show that in fact the equality holds. Any conjugacy class in PSL(2, Z) of infinite order has a representative of the form Moreover, such autoequivalence is pseudo-Anosov in the sense of [19], by [ h pol t (T 1 ) = h pol t (T 2 ) = 0 for t < 0, and since T 1 and T 2 are parabolic, we have h pol 0 (T 1 ) = h pol 0 (T 2 ) = 1. Hence the categorical polynomial entropy functions of T 1 and T 2 are discontinuous at t = 0. 6.5. P-twists. As an analogue of spherical objects, Huybrechts-Thomas [28] studied the so-called P-objects. Recall that given a saturated triangulated category D endowed with a Serre functor S, an object E in D is called a P d -object for some positive integer d, if S(E) ∼ = E[2d] and Hom D (E, E[ * ]) ∼ = H * (P d C , Z) ⊗ k as k-algebras. Examples of P-objects include line bundles in the derived category of a projective hyper-Kähler manifold, the structure sheaf of an embedded P d inside a 2d-dimensional holomorphic symplectic variety, etc.
Similar to spherical twits, any P-object E also induces an autoequivalence P E of the triangulated category [28], called P-twists. We refer to [28] for the precise definition and properties. Note that As an analogue of Lemma 6.12, we have the following estimate, which improves upon [18, Proof of Theorem 3.1]. Lemma 6.17 (Upper bound). For any objects G, G ′ ∈ D and any positive integer n, we have Proof. The proof is similar to Lemma 6.12. Applying P n−1 E to the distinguished triangle Therefore, ǫ t (G ′ , P n E (G)) ≤ ǫ t (G ′ , P n−1 E (G)) + A t e (1−2d(n−1))t , hence The case t = 0 follows immediately. For t = 0, we get ǫ t (G ′ , P n E (G)) ≤ B t + A t e t e −2dt − 1 (e −2dtn − 1).
One can conclude easily by separating the cases t > 0 and t < 0.
Proposition 6.18. Let d be a positive integer. Let E be a P d -object in a saturated triangulated category D. Then (i) If t < 0, then h pol t (P E ) = 0; (ii) If t > 0 and E ⊥ = 0, then h pol t (P E ) = 0; (iii) If t = 0, then 0 ≤ h pol (P E ) ≤ 1.
Proof. Once we have Lemma 6.17, the proof goes exactly as in Proposition 6.13. We leave the details to the reader. 6.6. Autoequivalences for curves. In this section, we study categorical polynomial entropy of autoequivalences of derived categories of smooth projective curves. The discussion splits into two parts: standard autoequivalences (which covers the cases of non-elliptic curves), and elliptic curves (where the Fourier-Mukai transform plays an essential role). 6.6.1. Standard autoequivalences. Since Aut(E) ⋉ (Pic 0 (E) × Z [2]) is a normal subgroup in Aut(D b (E)), for each n there exists g n ∈ Aut(E) ⋉ (Pic 0 (E) × Z [2]) such that F n = F ′n g n . = δ(G, F ′n G)δ(F ′n G, F ′n g n G ∨ ) ≤ δ(G, F ′n G)δ(G, g n G ∨ ) By [32, Lemma 3.4], we have h cat (F ) = h cat (F ′ ). Therefore, (6.4) log δ(G, F n G ∨ ) − nh cat (F ) log n ≤ log δ(G, F ′n G) − nh cat (F ′ ) log n + log δ(G, g n G ∨ ) log n .
Taking limit n → ∞ of (6.4) gives h pol (F ) ≤ h pol (F ′ ). One can prove h pol (F ′ ) ≤ h pol (F ) using the same argument. Hence we have h pol (F ) = h pol (F ′ ).
The main result of this section is the following trichotomy, where the statements concerning categorical entropy is due to Kikuta [32, Section 3.2]. One sees clearly how polynomial entropy further refines his study. Since χ(G ⊗ F n G) = Aρ(N (F )) n + Bρ(N (F )) −n for some constant A, B with A = 0, we obtain that h pol (F ) = 0. Moreover, in this case N (F ) has eigenvalues ρ(N (F )) > 1 and 1 > ρ(N (F )) −1 > 0. Each of the eigenvalues has a single Jordan block of size one. Hence h pol (F ) = s(N (F )) = 0.