2.1 Lagrangian Floer theory

Let $\hat {W}$ be a Weinstein manifold with a Liouville one form $\lambda $ , which means that $(\hat {W},\omega =d\lambda )$ is a non-compact symplectic manifold such that the Liouville vector field Z, that is, Z is a vector field on $\hat {W}$ characterized by $\iota _Z \omega = \lambda $ , is gradient-like with respect to a certain Morse function on $\hat {W}$ . See [Reference Cieliebak and Eliashberg15] for a reference on Weinstein manifolds. Then, there exists a Weinstein domain W whose completion is $\hat {W}$ . In other words, $(W, \omega |_{W}=d\lambda |_{W})$ is a compact symplectic submanifold with boundary of $\hat {W}$ of codimension $0$ , for which the Liouville vector field points outward along the boundary $\partial W$ , and $\hat {W}$ is obtained from W by gluing the cylindrical end $[1,\infty ) \times \partial W$ along $\partial W$ , namely,

$$\begin{align*}\hat{W}= W \cup \left([1, \infty)\times \partial W \right).\end{align*}$$

Furthermore, on the cylindrical end $[1,\infty ) \times \partial W$ , the Liouville one form $\lambda $ satisfies

$$\begin{align*}\lambda|_{[1,\infty) \times \partial W} := r \alpha\end{align*}$$

where r is the coordinate for $[1,\infty )$ and $\alpha := \lambda |_{\partial W}$ . We remark that $\partial W$ is a contact manifold with the contact form $\alpha $ . Furthermore, since the Liouville vector field Z does not vanish outside a compact subset of $\hat {W}$ , we may consider the orbit space $\partial _\infty \hat {W}$ of the flow of Z at infinity, which we will call the ideal boundary of $\hat {W}$ . One can deduce that $\partial W$ is naturally diffeomorphic to $\partial _\infty \hat {W}$ .

To discuss the Floer theory of a pair of Lagrangian submanifolds of $\hat {W}$ , let J be an almost complex structure on $\hat {W}$ that is compatible with the symplectic form $\omega = d \lambda $ and is of contact type at $\infty $ . The first condition means that the symplectic structure $\omega $ and the almost complex structure J determine a Riemannian metric g on W given by

$$\begin{align*}g(\cdot ,\cdot ) = \omega( \cdot, J \cdot).\end{align*}$$

The latter condition means that the almost complex structure J maps the Liouville vector field Z to the Reeb vector field associated with $\alpha $ on $\partial W$ and preserves the contact distribution, which is necessary to apply maximum principles to J-holomorphic curves in $\hat {W}$ .

Definition 2.2. An exact Lagrangian submanifold L of $(\hat {W},\omega )$ is cylindrical (at $\infty $ ) if L and $\partial W$ intersect transversely and

$$\begin{align*}L \cap \left([1,\infty) \times \partial W \right) = [1,\infty)\times \mathcal{L},\end{align*}$$

where $\mathcal {L} := L \cap \partial W$ .

For convenience, we will use the term “Lagrangian” instead of “exact Lagrangian submanifold that is cylindrical at $\infty $ ”.

Let us assume that $L_1$ and $L_2$ are a transversal pair of Lagrangians in $\hat {W}$ . Since $L_i$ is an exact Lagrangian submanifold, there is a primitive function

$$\begin{align*}h_i : L_i \to \mathbb{R},\end{align*}$$

such that $\lambda |_{L_i} = d f_i$ . Let us fix such a primitive function $h_i$ for each Lagrangian $L_i$ .

Let $\mathcal {P}(L_1, L_2)$ be the space of paths from $L_1$ to $L_2$ in $\hat {W}$ . We define the action functional

(2.1) $$ \begin{align} \mathcal{A} = \mathcal{A}_{L_1, L_2} : \mathcal{P}(L_1, L_2) \to \mathbb{R} \end{align} $$

by

$$\begin{align*}\mathcal{A}(\gamma) := -\int \gamma^*\lambda - h_1(\gamma(0)) + h_2(\gamma(1)), \forall \gamma \in \mathcal{P}(L_1,L_2).\end{align*}$$

Let us further equip the path space $\mathcal {P}(L_1,L_2)$ with the standard $L^2$ -metric induced by g. Then it is straightforward to check that

• • critical points of the action functional $\mathcal {A}$ are constant paths from $L_1$ to $L_2$ , that is, intersection points of $L_1$ and $L_2$ , and

• • gradient flows are given by strips

  $$\begin{align*}u : \mathbb{R} \times [0,1] \to \hat{W}\end{align*}$$
  satisfying
  $$\begin{align*}u(s,0) \in L_1, u(s,1)\in L_2, \forall s\in \mathbb{R}\end{align*}$$
  and the J-holomorphic equation
  $$ \begin{align*} \partial_s u+ J \partial_t u =0& , \forall (s,t) \in \mathbb{R} \times [0,1]. \end{align*} $$

The Lagrangian Floer complex $CF^*(L_1,L_2)$ is given by the Morse complex for the action functional $\mathcal {A} =\mathcal {A}_{L_1,L_2}$ . Indeed, for a given field k,

• • $CF^*(L_1,L_2)$ is a graded k-vector space generated by intersection points $L_1 \cap L_2$ , and

• • the differential $\delta : CF^*(L_1,L_2) \to CF^{*+1} (L_1,L_2)$ is defined by counting J-holomorphic strips of index $1$ between two intersection points of $L_1$ and $L_2$ .

We call the resulting cohomology Lagrangian Floer cohomology between $L_1$ and $L_2$ and denote it by $HF^*(L_1,L_2)$ .

We would like to point out that the grading of the Floer cochain complex is not crucial in our remaining arguments. We will just assume that one of the followings holds.

• • either the Floer complex $CF^*(L_1,L_2)$ is $\mathbb {Z}/2$ -graded, which is always possible, or

• • it is $\mathbb {Z}$ -graded assuming that

  $$\begin{align*}2 c_1(\hat{W}) =0\end{align*}$$
  and the Lagrangians $L_1$ and $L_2$ are graded in the sense of [Reference Seidel45, Section 12].

Later in Section 7 where the grading is not important, we will usually denote the Floer cochain complex just by $CF(L_1,L_2)$ without the superscript $^*$ .

We remark that one can extend the action functional $\mathcal {A}$ (2.1) as a function defined on $CF^*(L_1,L_2)$ as

(2.2) $$ \begin{align} \mathcal{A}\left(\sum_{x_i \in L_1 \cap L_2} a_i x_i\right) = \max\left\{\mathcal{A}(x_i) | a_i \neq 0\right\}. \end{align} $$

We need the extended action functional $\mathcal {A}$ in Section 7.

2.2 Partially wrapped Fukaya category

Let W be a Weinstein domain and let $(\hat {W},\omega =d\lambda )$ be the completion of W as in the previous subsection. The purpose of this subsection is to provide a brief introduction of partially wrapped Fukaya category associated with a stop of $\hat {W}$ , mainly focusing on the definition of morphism space. There are two good references [Reference Ganatra, Pardon and Shende29], [Reference Sylvan47] for this topic. For convenience, we stick to the description given in [Reference Sylvan47]. A stop $\Lambda $ of $\hat {W}$ is a (possibly empty) hypersurface with boundary of $\partial W$ (or equivalently $\partial _{\infty } \hat {W}$ ) such that $\Lambda $ becomes a Liouville domain with the Liouville form $\lambda |_{\Lambda }$ .

Let $\Lambda \subset \partial W= \partial _{\infty } \hat {W}$ be a stop. The partially wrapped Fukaya category $\mathcal {W}(W,\Lambda )$ is an $A_{\infty }$ -category associated with the pair $(W,\Lambda )$ whose objects are cylindrical Lagrangians L of $\hat {W}$ equipped with additional structures such as gradings, relative pin structures and local systems such that $L \cap \partial W \cap \Lambda =\emptyset $ .

To define morphism spaces of $\mathcal {W}(W,\Lambda )$ , we consider quadratic Hamiltonians H on $\hat {W}$ that are compatible with the stop $\Lambda $ in the sense of [Reference Sylvan47, Section 2.4]. Let us denote the set of all such Hamiltonians on $\hat {W}$ by $\mathcal {H}(\hat {W})$ . For any Hamiltonian H on $\hat {W}$ , we denote the corresponding Hamiltonian vector field by $X_H$ , that is, $X_H$ is a unique vector field on $\hat {W}$ characterized by

$$\begin{align*}\iota_{X_H}\omega = -dH.\end{align*}$$

We further consider an almost complex structure J that is compatible with $\omega $ and is of contact type as in the previous subsection.

Let $L_1$ and $L_2$ be Lagrangians that are allowed to be objects of $\mathcal {W}(W,\Lambda )$ . For a Hamiltonian $H \in \mathcal {H}(\hat {W})$ , we define an action functional

(2.3) $$ \begin{align} \mathcal{A}_{L_1,L_2,H} : \mathcal{P}(L_1,L_2) \to \mathbb{R} \end{align} $$

by

$$\begin{align*}\mathcal{A}_{L_1,L_2,H} (\gamma) = -\int \gamma^*\lambda + \int_{0}^{1} H(\gamma(t))dt- h_1(\gamma(0)) + h_2(\gamma(1)).\end{align*}$$

As done in the previous subsection, it is easy to check that

• • critical points of the action functional $\mathcal {A}_{L_1,L_2,H}$ are $X_H$ -chords from $L_1$ to $H_2$ , where an $X_H$ -chord from $L_1$ to $L_2$ means a path $\gamma :[0,1]\to \hat {W}$ satisfying $\gamma (0)\in L_1$ , $\gamma (1) \in L_2$ and $\dot {\gamma }(t) = X_H(\gamma (t))$ , and

• • gradient flows are given by Floer strips

  $$\begin{align*}u : \mathbb{R} \times [0,1] \to \hat{W}\end{align*}$$
  satisfying
  $$\begin{align*}u(s,0) \in L_1, u(s,1)\in L_2, \forall s\in \mathbb{R}\end{align*}$$
  and the Floer equation
  $$\begin{align*}\partial_s u+ J (\partial_t u - X_H(u)) =0 , \forall (s,t) \in \mathbb{R} \times [0,1].\end{align*}$$

Let us denote by $\mathcal {X}(L_1,L_2;H)$ the set of all $X_H$ -chords from $L_1$ to $L_2$ . There is a function $n_{\Lambda } : \mathcal {X}(L_1,L_2;H) \to \mathbb {Z}_{\geq 0}$ defined by counting the intersection number of a given $X_H$ -chord with the stop $\Lambda $ . We take $\mathcal {X}_0(L_1,L_2;H)$ to be $n_{\Lambda }^{-1}(0)$ .

For a given field k, the partially wrapped Floer complex $CW_{\Lambda }^*(L_1,L_2)$ is defined as follows.

• • $CW^*_{\Lambda }(L_1,L_2)$ is a graded k-vector space generated by elements of $\mathcal {X}_0(L_1,L_2;H)$ , and

• • the differential is defined by counting Floer strips of index $1$ between two elements of $\mathcal {X}_0(L_1,L_2;H)$ .

It can be shown as in [Reference Sylvan47] that this indeed defines a cochain complex. We call the resulting cohomology the partially wrapped Floer cohomology between $L_1$ and $L_2$ with respect to the stop $\Lambda $ and denote it by $HW^*_{\Lambda }(L_1,L_2)$ .

There are two particularly interesting cases of a stop. The first one is the case $\Lambda =\emptyset $ . In this case, the corresponding $CW^*_{\Lambda }(L_1,L_2)$ (resp. $HW^*_{\Lambda }(L_1,L_2)$ ) is called the wrapped Floer complex between $L_1$ and $L_2$ (resp. wrapped Floer cohomology between $L_1$ and $L_2$ .) and denoted by $CW^*(L_1,L_2)$ (resp. $HW^*(L_1,L_2)$ ).

The second one is the case when the partially wrapped Floer complex (or more generally, partially wrapped Fukaya category) is fully stopped. To be more precise, we define the notion of full-stop.

We would like to note two things. The first one is the reason why the notion of full-stop is interesting. The reason is actually mentioned in Definition 2.3, that is, the morphism spaces have finite dimensional homologies. Equivalently, we can simply say that the corresponding partially wrapped Fukaya category is proper. This algebraic property plays a key role in the current article. Then, a question naturally arises, which is the second one we would like to note. The question asks the existence of full-stops. We note that for every Weinstein manifold, the existence is guaranteed. For example, see [Reference Ganatra, Pardon and Shende29, Section 8.6].

To complete an introduction of partially wrapped Fukaya category, we need to explain the $A_{\infty }$ -operations on it. Indeed, for a positive integer k, we call a disk with $(k+1)$ -boundary punctures a $(k+1)$ -gon. We further label the boundary punctures with $\{1,\dots ,k+1 \ \mathrm {mod} \ k+2\}$ and call the boundary component between ith and $(i+1)$ -puncture the ith-boundary component for $1\leq i \leq k+1$ . For any collection $(L_1,\dots , L_{k+1})$ of Lagragians of W, the $A_\infty $ -operation

$$\begin{align*}\mu^k_{\Lambda} : CW^*_{\Lambda}(L_k,L_{k+1})\otimes \dots \otimes CW^*_{\Lambda}(L_1,L_2) \to CW^*_{\Lambda}(L_1,L_{k+1})\end{align*}$$

is defined by counting rigid Floer $(k+1)$ -gons, which map the ith boundary component to $L_i$ and are asymptotic to an $X_H$ -chord from $L_i$ to $L_{i+1}$ belonging to $\mathcal {X}_0(L_i,L_{i+1};H)$ at the ith puncture for $1\leq i \leq k$ and an $X_H$ -chord from $L_1$ to $L_{k+1}$ at $k+1$ -th puncture. Here $L_{k+2}$ is defined to be $L_1$ for simplicity. Especially the first operation $\mu ^1_{\Lambda }$ coincides with the differential on the partially wrapped Floer complex up to sign. Then the operations $\mu ^k_{\Lambda }$ ’s satisfy the $A_{\infty }$ -relations as shown in [Reference Sylvan47].

The resulting $A_{\infty }$ -category is the partially wrapped Fukaya category $\mathcal {W}(W,\Lambda )$ of the pair $(W,\Lambda )$ . As done above, in the case $\Lambda = \emptyset $ , the corresponding partially wrapped Fukaya category is called the wrapped Fukaya category of W and denoted by $\mathcal {W}(W)$ . For future use in this paper, we further define the compact Fukaya category $\mathcal {F}(W)$ to be the full subcategory of $\mathcal {W}(W)$ generated by closed exact Lagrangian submanifolds of $\hat {W}$ .

2.3 Topological entropy

Let X be a topological space and let $\phi : X \to X$ be a continuous self-mapping defined on X. The notion of topological entropy $h_{top}(\phi )$ is defined in [Reference Adler, Konheim and McAndrew2] for compact X and in [Reference Hofer32], [Reference Hofer33] for non-compact X.

In particular, even though X is non-compact, if a continuous self-mapping $\phi : X \to X$ is compactly-supported and so its support is contained in a compact subspace $X_0$ of X, then the following equality holds

$$\begin{align*}h_{top} (\phi|_{X_0}) = h_{top}(\phi).\end{align*}$$

Since we only consider the topological entropy of compactly-supported smooth self-mappings on smooth manifolds in this paper, the above observation implies that it is enough to consider the topological entropy of a smooth self-mapping on a compact smooth manifold. Let X be a compact smooth manifold (with or without boundary) of dimension n equipped with a Riemannian metric g in the rest of Section 2.3. Then the following is a formulation of the topological entropy of a smooth map $\phi : X \to X$ due to Dinaburg [Reference Dinaburg19] and Bowen [Reference Bowen11].

Note that the $h_{top}(\phi )$ in Definition 2.4 (4) does not depend on a specific choice of a Riemannian metric g. See [Reference Bowen11], [Reference Dinaburg20] for more details on topological entropy.

We end this subsection by stating a property of topological entropy that plays a key role in the proof of Lemma 3.4. For a $C^\infty $ -submanifold $Y \subset X$ of dimension m, let

$$\begin{align*}\Gamma_{\phi|_Y}^k := \Big\{\left(y, \phi(y), \dots, \phi^{k-1}(y)\right) \in X^k | y \in Y \Big\}.\end{align*}$$

We note that the product metric on $X^k$ induces an m-dimensional volume form. Thus, we can measure the volumes of $\Gamma _{\phi |_Y}^k$ for all k. The following is a significant observation due to Yomdin that the exponential growth rate of the volumes is a lower bound of $h_{top}(\phi )$ .

Theorem 2.5 (Yomdin).

The topological entropy of $\phi $ is bounded by the exponential growth rate of the volume of $\Gamma _{\phi |_Y}^k$ , that is,

$$\begin{align*}\limsup_{k \to \infty} \frac{1}{k} \log \mathrm{Vol}(\Gamma_{\phi|_Y}^k) \leq h_{top}(\phi).\end{align*}$$

See [Reference Gromov30], [Reference Yomdin50] for the proof of Theorem 2.5.

2.4 Categorical entropy

In this subsection, we introduce the notion of categorical entropy. This requires preliminary knowledge on triangulated categories, split-generators, etc. We recommend [Reference Gelfand and Manin27], [Reference Seidel45] as a general reference on these. We start by introducing the notions of complexity and categorical entropy, which are originally defined in [Reference Dimitrov, Haiden, Katzarkov and Kontsevich18].

In the current paper, we only consider the case of $t=0$ .

Definition 2.7. Let $\Phi : \mathcal {C} \to \mathcal {C}$ be an auto-equivalence defined on a triangulated category $\mathcal {C}$ with a generator G. We define the categorical entropy of $\Phi $ as

$$\begin{align*}h_{cat}(\Phi):= h_{cat}(\Phi;0).\end{align*}$$

Let $\mathcal {D}$ be a fully faithful subcategory of $\mathcal {C}$ such that

• • $\mathcal {D}$ is a triangulated category, and

• • the restriction of $\Phi $ to $\mathcal {D}$ defines an auto-equivalence on $\mathcal {D}$ , that is, $\Phi (\mathcal {D}) \subset \mathcal {D}$ .

It is known that there exists a localization functor l

$$\begin{align*}l : \mathcal{C} \to \mathcal{C}/\mathcal{D}.\end{align*}$$

See [Reference Drinfeld21]. Then, $\Phi $ induces an auto-equivalence defined on $\mathcal {C}/\mathcal {D}$ uniquely up to natural transformations. More precisely, there exists a unique (up to natural transformation) dg functor

$$\begin{align*}\Phi_{\mathcal{C}/\mathcal{D}}: \mathcal{C}/\mathcal{D}\to\mathcal{C}/\mathcal{D},\end{align*}$$

satisfying

$$\begin{align*}\Phi_{\mathcal{C}/\mathcal{D}}\circ l=l\circ \Phi.\end{align*}$$

To be clear, let us use the following notations $\Phi _{\mathcal {C}}, \Phi _{\mathcal {D}}$ , and $\Phi _{\mathcal {C}/\mathcal {D}}$ ,

$$\begin{align*}\Phi_{\mathcal{C}}:= \Phi:\mathcal{C} \to \mathcal{C}, \Phi_{\mathcal{D}}:= \Phi_{\mathcal{C}}|_{\mathcal{D}}: \mathcal{D} \to \mathcal{D}, \text{ and } \Phi_{\mathcal{C}/\mathcal{D}}: \mathcal{C}/\mathcal{D} \to \mathcal{C}/\mathcal{D}.\end{align*}$$

Then, [Reference Bae, Choa, Jeong, Karabas and Lee9, Theorem 3.8] compares the categorical entropies of $\Phi _{\mathcal {C}}, \Phi _{\mathcal {D}}$ , and $\Phi _{\mathcal {C}/\mathcal {D}}$ .

Lemma 2.9 [Reference Bae, Choa, Jeong, Karabas and Lee9, Theorem 3.10].

The categorical entropies of $\Phi _{\mathcal {C}}, \Phi _{\mathcal {D}}, \Phi _{\mathcal {C}/\mathcal {D}}$ satisfy

$$\begin{align*}h_{cat}(\Phi_{\mathcal{C}/\mathcal{D}}) \leq h_{cat}(\Phi_{\mathcal{C}}) \leq \max \{h_{cat}(\Phi_{\mathcal{D}}), h_{cat}(\Phi_{\mathcal{C}/\mathcal{D}})\}.\end{align*}$$

Let $\hat {W}$ be a Weinstein manifold. Then let $W \subset \hat {W}$ be an associated Weinstein domain and let $\Lambda $ be a stop in $\partial W=\partial _\infty \hat {W}$ . Let $\phi : \hat {W} \to \hat {W}$ be a compactly supported exact symplectic automorphism and. Then $\phi $ induces functors $\Phi : \mathcal {W}(W) \to \mathcal {W}(W)$ and $\Phi _\Lambda : \mathcal {W}(W,\Lambda ) \to \mathcal {W}(W,\Lambda )$ . Thanks to Lemma 2.9, one can compare $h_{cat}(\Phi )$ and $h_{cat}(\Phi _\Lambda )$ .

Lemma 2.10 [Reference Bae, Choa, Jeong, Karabas and Lee9, Theorem 1.2].

The induced functors $\Phi $ and $\Phi _\Lambda $ have the same categorical entropy, that is,

$$\begin{align*}h_{cat}(\Phi) = h_{cat}(\Phi_\Lambda).\end{align*}$$

Proof. We note that

$$\begin{align*}\mathcal{W}(W) := \mathcal{W}(W,\Lambda)/\mathcal{D},\end{align*}$$

where $\mathcal {D}$ means the full subcategory of $\mathcal {W}(W,\Lambda )$ generated by all linking disks. Here linking disks are certain cylindrical Lagrangian disks of $\hat {W}$ associated with $\Lambda $ , for which the intersection with the domain W can be chosen to lie arbitrarily close to $\Lambda \subset \partial W$ . See [Reference Ganatra, Pardon and Shende29, Section 5] for a definition of linking disk.

Since $\phi $ is compactly supported, the restriction of $\Phi $ on $\mathcal {D}$ is the identity functor. Thus, the categorical entropy of $\Phi |_{\mathcal {D}}$ is zero.

We note that, as mentioned Remark 2.8,

$$\begin{align*}h_{cat}(\Phi), h_{cat}(\Phi_\Lambda) \geq 0.\end{align*}$$

By applying Lemma 2.9, one has

$$\begin{align*}0 \leq h_{cat}(\Phi_\Lambda) \leq h_{cat}(\Phi) \leq \max\{h_{cat}(\Phi_\Lambda), 0\} = h_{cat}(\Phi_\Lambda).\end{align*}$$

This completes the proof.

3.1 Lagrangian tomograph

Let $(\hat {W},d\lambda )$ be a Weinstein manifold and let $W \subset \hat {W}$ be a Weinstein domain in the rest of this section. In many places of this paper, we consider pairs of Lagrangians in $\hat {W}$ satisfying the following condition.

Definition 3.1. A pair of Lagrangians $(L_1, L_2)$ in $\hat {W}$ is good if $L_1$ and $L_2$ are disjoint in the cylindrical part, that is,

$$\begin{align*}L_1 \cap L_2 \cap \left([1,\infty) \times \partial W \right) = \varnothing.\end{align*}$$

For a good pair of Lagrangians, we construct a Lagrangian tomograph in Lemma 3.2. The original construction of Lagrangian tomograph is given in [Reference Cineli, Ginzburg and Gürel13, Section 5.2.3], and our construction is a slight modification of the original one.

We note that the radius of the ball $B_\epsilon ^d$ in the above lemma is $\delta $ , not $\epsilon $ . The reason why we use the notation is explained in Remark 3.3.

Before going further, we briefly review the notion of Hofer norm $d_H$ of a Hamiltonian isotopy, which appears in the condition (ii) of Lemma 3.2. Let $\varphi $ be a compactly supported Hamiltonian isotopy. Then, the Hofer norm of $\varphi $ is defined as

$$\begin{align*}\lVert \varphi \rVert_{Hofer} := \inf_H \int_{S^1} (\max_M H_t - \min_M H_t)dt,\end{align*}$$

where the infimum is taken over all $1$ -periodic, time-dependent Hamiltonian H generating $\varphi $ . Moreover, one can define the Hofer distance between two Hamiltonian isotopic Lagrangians L and $L'$ as

$$\begin{align*}d_H(L,L'):= \inf\{\lVert \varphi \rVert_{Hofer} | \varphi(L)= L'\}.\end{align*}$$

Proof of Lemma 3.2.

Since $(L_1,L_2)$ is a good pair, there is a submanifold $W_0 \subset W$ of codimension 0 whose closure is compact such that

$$\begin{align*}L_1 \cap L_2 \subset \mathrm{Int}(W_0) \subset W_0 \subset \mathrm{Int}(W),\end{align*}$$

where $\mathrm {Int}(W_0)$ and $\mathrm {Int}(W)$ denote the interiors of $W_0$ and W, respectively. Then, we choose a collection of real-valued functions

$$\begin{align*}\{g_1, \dots, g_d | g_i: L_1 \to \mathbb{R}\},\end{align*}$$

satisfying

1. (A) $g_i(x)=0$ if $x \in L_1 \setminus W$ , and

2. (B) for all $x \in L_1 \cap W_0$ , the cotangent fiber $T^*_xL_1$ is generated by $\{dg_i(x) | i =1, \dots , d\}$ .

For any $s = (s_1, \dots , s_d) \in \mathbb {R}^d$ , We set

$$ \begin{align*} & f_s: L_1 \to \mathbb{R}, \\ x \mapsto s_1 g_1&(x) + \dots + s_d g_d(x). \end{align*} $$

We note that there is a small neighborhood U of $L_1$ in $\hat {W}$ , which is symplectomorphic to a small disk cotangent bundle of $L_1$ . We call U a Weinstein neighborhood of $L_1$ . Then for $s \in \mathbb {R}^d$ such that $\lVert s \rVert \ll 1$ , one can assume that the graph of $df_s$ is embedded into $\hat {W}$ . Let $L^s$ be the embedded image of the graph of $df_s$ in $\hat {W}$ . By the construction of $L^s$ , (i) holds obviously.

Then, one can observe that (ii) holds for all $s \in \mathbb {R}^d$ sufficiently close to the origin. Let $\delta>0$ be a sufficiently small number such that if $\lVert s \rVert < \delta $ , then (ii) holds. Let $B_\epsilon ^d \subset \mathbb {R}^d$ be the ball of radius $\delta $ centered at the origin.

In order to prove (iii), we show that the following map $\Psi $ is submersive at every point of $\Psi ^{-1}(L_2)$ :

(3.1) $$ \begin{align} & \Psi: B^d_\epsilon \times L_1 \to \hat{W}, \\ & \notag (s,x) \mapsto df_s(x). \end{align} $$

In other words, if $\Psi (s,x) \in L_2$ for some $(s,x) \in B^d_\epsilon \times L_1$ , we show that

$$\begin{align*}D\Psi_{(s,x)}: T_{(s,x)}\left(B^d_\epsilon \times L_1\right) \simeq T_s B^d_\epsilon \oplus T_x L_1 \to T_{\Psi(s,x)}\hat{W}\end{align*}$$

is surjective. Then it will follow that $L^s$ and $L_2$ intersect transversely for almost all $s\in B_\epsilon ^d$ , that is, (iii) holds.

We equip $\hat {W}$ with a Riemannian metric g compatible with the Weinstein neighborhood of $L_1$ in the following sense. Indeed, note that any Riemannian metric $g_{L_1}$ on $L_1$ determines a natural Riemannian metric on its contangent bundle $T^*L_1$ . Then identifying the Weinstein neighborhood U with a disk subbundle of $T^*L_1$ , we require that the restriction of g to U coincides with the unique natural Riemmanian metric induced by the Riemannian metric $g_{L_1} := g|_{L_1}$ in the above sense. This is always possible.

Now let

(3.2) $$ \begin{align} \ell := \min\left\{ d(x,y)| x \in L_1 \cap \left(W \setminus \mathrm{Int}(W_0)\right), y \in L_2 \cap \left(W \setminus \mathrm{Int}(W_0)\right)\right\}, \end{align} $$

where $d(x,y)$ is the distance function with respect to the Riemannian metric g. Since both $L_1\cap (W\setminus \mathrm {Int}(W_0))$ and $L_2\cap (W\setminus \mathrm {Int}(W_0))$ are compact and $L_1 \cap L_2 \subset \mathrm {Int}(W_0)$ , $\ell $ is well-defined and positive.

Consider the Riemannian metric on $L_1$ given by the restriction of g. Let us denote by $\lVert \cdot \rVert $ the corresponding norm on $T^*_xL_1$ . If the radius $\delta $ is sufficiently small, then $\lVert df_s(x) \rVert < \ell $ for all $(s,x) \in B^d_\epsilon \times W$ since $g_i$ ’s are compactly supported. We assume that $\delta $ is sufficiently small in this sense in the rest of the proof.

Now assume that $\Psi (s,x) \in L_2$ for some $(s,x)\in B^{d}_\epsilon \times L_1$ . If $\Psi (s,x) \in \hat {W} \setminus W$ , then $\Psi (s,x) \in L_1$ since $g_i$ ’s are assumed to be zero over $\hat {W} \setminus W$ by (A). This contradicts to the assumption $L_1 \cap L_2 \subset W_0 \subset W$ . Otherwise, if $\Psi (s,x) \in W \setminus \mathrm {Int}(W_0)$ , then $d(\Psi (s,x),x) = \lVert df_s(x) \rVert < \ell $ , which follows from our choice of the Riemannian metric g. But, this contradicts to Equation (3.2).

The above paragraph shows that if $\Psi (s,x) \in L_2$ , then $\Psi (s,x) \in W_0$ , that is, $L^s \cap L_2 \subset W_0$ . By the assumption (B), this means that $\Psi $ is submersive at every point in $\Psi ^{-1}(L_2)$ .

3.2 Crofton’s inequality

In Section 3.2, we prove Lemma 3.4, that is, a Crofton type inequality, which plays a key role in the proof of Theorem 1.2.

In order to state Lemma 3.4, we need some preparation. For $s \in B^d_\epsilon $ such that $L^s \pitchfork L_2$ , let

$$\begin{align*}N(s) := |L^s \cap L_2|.\end{align*}$$

Then, $N(s)$ is finite for almost all $s \in B^d_\epsilon $ . Moreover, $N(s)$ is an integrable function on $B^d_\epsilon $ .

Since $B^d_\epsilon \subset \mathbb {R}^d$ , $B^d_\epsilon $ carries the standard Euclidean metric. Let $ds$ be the volume form on $B^d_\epsilon $ induced by the Euclidean metric.

Let

$$\begin{align*}E := \Psi^{-1}(W_0).\end{align*}$$

Then, let us fix a metric $g_E$ on E such that the restriction of $D\Psi $ to the normals to $\Psi ^{-1}(y), y \in W$ is an isometry. Since $\Psi $ is a proper submersion, $\Psi $ is a locally trivial fibration by Ehresmann’s fibration theorem [Reference Ehresmann22]. Thus, the existence of such a metric is guaranteed.

Now, we state Lemma 3.4.

Lemma 3.4. The following inequality holds:

(3.3) $$ \begin{align} \int_{B^d_\epsilon} N(s) ds \leq C \cdot \mathrm{Vol}(L_2 \cap W), \end{align} $$

where C is a constant depending only on $\Psi , ds$ , the fixed metric g on $\hat {W}$ , and the fixed metric $g_E$ on E.

Proof. Let $\Sigma := \Psi ^{-1}(L_2 \cap W)$ . Then, by definition, for all $s \in B_{\epsilon }^d$ such that $L^s \pitchfork L_2$ , one has

$$\begin{align*}|(s \times L_1) \cap \Sigma| = |L^s \cap L_2| = N(s).\end{align*}$$

Note that in the proof of Lemma 3.2, we have

$$\begin{align*}L^s \cap L_2 \subset W_0 \subset W,\end{align*}$$

by choosing a sufficiently small $B^d_\epsilon $ .

We recall that $B^d_\epsilon $ carries the Euclidean metric and $L_1$ also carries a metric $g|_{L_1}$ . Thus, $B^d_\epsilon \times L_1$ carries a product metric. On E, the restriction of the product metric gives another metric that does not need to be the same as $g_E$ .

Let $\pi : E \hookrightarrow B^d_\epsilon \times L_1 \to B^d_\epsilon $ be the projection to the first factor. Then, if $\mathrm {Vol}_1(\cdot )$ denotes the volume with respect to the product metric on E, one has

(3.4) $$ \begin{align} \int_{B^d_\epsilon} N(s) ds = \int_{B^d_\epsilon} |(s \times L_1) \cap \Sigma |ds = \int_{\Sigma} \pi^*ds \leq \mathrm{Vol}_1(\Sigma). \end{align} $$

Let $\mathrm {Vol}(\cdot )$ (resp. $\mathrm {Vol}_2(\cdot )$ ) denote the volume with respect to the fixed metric g (resp. $g_E$ ) on W (resp. E). Then, by Fubini theorem, one has

(3.5) $$ \begin{align} \mathrm{Vol}_2(\Sigma) = \int_{L_2 \cap W} \mathrm{Vol}_2 \left(\Psi^{-1}(y)\right) dy|_{L_2} \leq \max_{y \in \Psi(E)} \mathrm{Vol}_2\left(\Psi^{-1}(y)\right) \cdot \mathrm{Vol}(L_2 \cap W). \end{align} $$

We note that since E is compact,

(3.6) $$ \begin{align} \mathrm{Vol}_1(\Sigma) \leq C_0 \cdot \mathrm{Vol}_2(\Sigma), \end{align} $$

where $C_0$ is a constant depending only on $g_E$ and the product metric on E.

By combining Equations 3.4–3.6, one concludes that

$$\begin{align*}\int_{B^d_\epsilon} N(s) ds \leq C \cdot \mathrm{Vol}(L_2 \cap W),\end{align*}$$

where C is a constant depending only on $\Psi , ds, g$ , and $g_E$ .

5.1 The first example

Let W be a 2-dimensional Weinstein domain such that $W \neq \mathbb {D}^2$ . It is well-known that its wrapped Fukaya category is generated by the Lagrangian cocores, see [Reference Chantraine, Rizell, Ghiggini and Golovko14], [Reference Ganatra, Pardon and Shende29]. This ensures that the categorical entropy of an endo-functor on $\mathcal {W}(W)$ is well-defined.

Let U be a small open ball in W. It is well-known that there is a Hamiltonian diffeomorphism $\widetilde {\phi }: \overline {U} \to \overline {U}$ defined on the closure of U such that

• • $\tilde {\phi }$ is the identity near the boundary of $\overline {U}$ , and

• • $\tilde {\phi }$ has a positive topological entropy.

Smale’s horseshoe map is an example of such $\tilde {\phi }$ .

Since $\widetilde {\phi }$ is assumed to be the identity near the boundary of the closure of U, it admits a trivial extension to the whole Weinstein domain W, which we will call $\phi $ . Since $\phi $ is a compactly supported Hamiltonian diffeomorphism, if we let $\Phi $ be its induced functor on $\mathcal {W}(W)$ as above, then we have

$$\begin{align*}h_{cat}(\Phi) = 0<h_{top}(\widetilde{\phi} = \phi|_{U}) \leq h_{top}(\phi).\end{align*}$$

Hence $\phi $ is an example showing that the inequality (1.1) can be strict.

5.2 The second example

Let $\phi :\hat {W} \to \hat {W}$ be a compactly supported exact symplectic automorphism of a Weinstein manifold $\hat {W}$ . Let $\phi _*: H_*(\hat {W}) \to H_*(\hat {W})$ be the linear map on the homology of $\hat {W}$ that $\phi $ induces. We define the spectral radius of $\phi $ as the maximal absolute value of eigenvalues of $\phi _*$ . Let $\mathrm {Rad}(\phi )$ denote the spectral radius of $\phi $ . It is a well-known fact due to Yomdin [Reference Yomdin50] that logarithm $\log \mathrm {Rad}(\phi )$ of the spectral radius is a lower bound of $h_{top}(\phi )$ . We refer the reader to [Reference Gromov30], [Reference Yomdin50] for more details. Since Theorem 4.1 gives another lower bound of $h_{top}(\phi )$ , that is, $h_{cat}(\Phi )$ , one can ask the relationship between two lower bounds of $h_{top}(\phi )$ . In this subsection, first, we give an example of $\phi $ such that

$$\begin{align*}\log \mathrm{Rad}(\phi) = 0 < h_{cat}(\phi).\end{align*}$$

The example shows that the categorical entropy could be an useful tool for proving positivity of topological entropy of symplectomorphisms. After introducing the example, we also introduce a construction of symplectomorphisms having positive topological entropy.

To introduce the example, let A and B be n-dimensional spheres. Then let $\hat {W}$ be the plumbing of the cotangent bundles $T^*A$ and $T^*B$ at a point. In other words, $\hat {W}$ is the Milnor fiber of $A_2$ -type.

Let $\tau _A$ and $\tau _B$ be the Dehn twist defined on W along A and B, respectively. Let us consider the symplectic automorphism on $\hat {W}$ given by

$$\begin{align*}\phi = \tau_A \circ \tau_B^{-1}.\end{align*}$$

Now observe that the homology of $\hat {W}$ is given by

$$\begin{align*}H_*(\hat{W}) = \begin{cases} \mathbb{Z} \langle [pt] \rangle &*=0, \\ \mathbb{Z} \langle [A], [B]\rangle & *=n,\\ 0 &\text{otherwise.} \end{cases}\end{align*}$$

Since $\phi $ induces the trivial map on the zeroth homology $H_0(\hat {W})$ , it is enough to consider its induced map on the n-dimensional homology $H_n(\hat {W})$ to compute $\mathrm {Rad}(\phi _*)$ .

For that purpose, we consider the induced map of $\tau _A$ and $\tau _B$ on $H_n(\hat {W})$ separately.

1. 1. $(\tau _A)_*([A]) = (-1)^{n-1}[A]$ .

2. 2. $(\tau _A)_*([B]) = [A]+[B]$ .

3. 3. $(\tau _B)_*([A]) = [A] +(-1)^n [B]$ .

4. 4. $(\tau _B)_*([B]) = (-1)^{n-1}[B]$ .

In other words, $(\tau _A)_*$ and $(\tau _B)_*$ are represented by the matrices

$$\begin{align*}(\tau_A)_* = \begin{pmatrix} (-1)^{n-1} & 1 \\ 0 & 1\end{pmatrix} \text{ and } (\tau_B)_* = \begin{pmatrix} 1& 0 \\ (-1)^n & (-1)^{n-1}\end{pmatrix}, \end{align*}$$

respectively.

Consequently, the map $\phi _* = (\tau _A)_* \circ (\tau _B^{-1})_*$ is represented by

$$ \begin{align*} \begin{pmatrix} (-1)^{n-1} & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1& 0 \\ (-1)^n & (-1)^{n-1}\end{pmatrix}^{-1} &= \begin{pmatrix} (-1)^{n-1} & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1& 0 \\ 1 & (-1)^{n-1}\end{pmatrix}\\ &= \begin{pmatrix}1+(-1)^{n-1}& (-1)^{n-1}\\ 1 & (-1)^{n-1} \end{pmatrix}. \end{align*} $$

Let us now assume that n is even. Then the above matrix is

$$\begin{align*}\begin{pmatrix} 0 & -1\\ 1 & -1 \end{pmatrix}. \end{align*}$$

A straightforward computation shows that its eigenvalues are

$$\begin{align*}\frac{-1 + \sqrt{3} i}{2} \text{ and } \frac{-1- \sqrt{3}i}{2}.\end{align*}$$

Hence the spectral radius of $\phi _*$ is

$$\begin{align*}\left|\frac{-1+\sqrt{3}i}{2} \right| = \left|\frac{-1- \sqrt{3}i}{2} \right| =1.\end{align*}$$

On the other hand, [Reference Bae and Lee10, Theorem 1.5] says that the categorical entropy of the induced map of $\phi $ on the wrapped Fukaya category is the same as its stretching factor. Moreover, one can easily compute the stretching factor of $\phi $ by employing the techniques in [Reference Bae and Lee10], and the stretching factor of the given example $\phi $ is

$$\begin{align*}\frac{3+\sqrt{5}}{2}.\end{align*}$$

We note that the detailed computation could be found in [Reference Bae and Lee10, Section 11.1].

Finally, we can prove that $\phi $ has positive topological entropy even though the spectral radius of $\phi $ is $1$ , since

$$\begin{align*}\log \mathrm{Rad}( \phi) =0 <\frac{3+\sqrt{5}}{2} = h_{cat} (\Phi) \leq h_{top}(\phi).\end{align*}$$

We would like to note that the above example $\phi $ is a symplectic automorphism constructed by a construction introduced in [Reference Lee37]. We introduce the construction given in [Reference Lee37] and prove that if a symplectic automorphism $\psi $ is compactly supported Hamiltonian isotopic to any symplectic automorphism $\phi $ constructed in [Reference Lee37], then $\psi $ has positive topological entropy.

The following is the construction given in [Reference Lee37] with a technical condition.

Definition 5.1. Let M be a symplectic manifold and $\phi :M \to M$ be a symplectic automorphism. We call that $\phi $ is of Penner type if there exist two collections of Lagrangian spheres in M,

$$\begin{align*}\{\alpha_1, \dots, \alpha_s\} \text{ and } \{\beta_1, \dots, \beta_t\}\end{align*}$$

satisfying the following conditions:

1. (i) $\phi $ is a product of positive powers of $\tau _i$ and negative powers of $\sigma _j$ where $\tau _i$ and $\sigma _j$ are generalized Dehn twists along $\alpha _i$ and $\beta _j$ , respectively.

2. (ii) $\{\alpha _1, \dots , \alpha _s\}$ and $\{\beta _1, \dots , \beta _t\}$ are pair-wisely disjoint collections of Lagrangian spheres and $\alpha _i$ and $\beta _j$ are transverse to each other.

3. (iii) Let G be a graph such that

   • • whose vertex set is $\{\alpha _1, \dots , \alpha _s, \beta _1, \dots , \beta _t\}$ and

   • • two vertices $v, w \in \{\alpha _1, \dots , \alpha _s, \beta _1, \dots , \beta _t\}$ are connected by k edges if the number of intersection points of two Lagrangian spheres is k, that is, $|v \cap w| =k$ .

   Then, the graph G is a tree.

Now, we prove the main result of the present subsection.

Theorem 5.3. Let M be a symplectic manifold of dimension $\geq 6$ . If a symplectic automorphism $\phi :M \to M$ is of Penner type, then

$$\begin{align*}h_{top}(\phi)> 0.\end{align*}$$

Proof. We note that any Penner type $\phi $ is compactly supported by Definition 5.1. Moreover, the compact support of $\phi $ is given as a small neighborhood of

$$\begin{align*}\cup \alpha_i \bigcup \cup \beta_j.\end{align*}$$

Thanks to Weinstein’s neighborhood theorem [Reference Weinstein49], we could assume that the compact support is a subset of a plumbing space P of $T^*S^n$ where n is the dimension of the Lagrangian spheres. Moreover, the plumbing pattern of P is determined by the intersection pattern of $\alpha _i$ and $\beta _j$ . Thus, the plumbing pattern is the graph G defined in 5.1 (iii).

Simply, we have

$$\begin{align*}\text{compact support of } \phi \subset P.\end{align*}$$

Now, we can consider a symplectic automorphism $\phi _0: P \to P$ , which is a natural extension of the restriction of $\phi $ on the compact support. Then, we have that the topological entropy of $\phi $ and $\phi _0$ coincide. Thus, it is enough to show that $h_{cat}(\Phi _0)>0$ .

Note that the domain of $\phi _0$ is a plumbing space P whose plumbing pattern is a tree. Thus, we can employ the results of [Reference Bae and Lee10]. Finally, [Reference Bae and Lee10, Theorems 1.5 and 7.1] proves that $h_{cat}(\Phi _0)>0$ .

7.1 Preliminaries

In this subsection, we review the theory of persistence modules, and we apply it to Lagrangian Floer homology. We refer the reader to [Reference Polterovich, Rosen, Samvelyan and Zhang42], [Reference Usher and Zhang48] for the theory of persistence modules. Also, we refer the reader to [Reference Cineli, Ginzburg and Gürel13] for the details we omitted in the current subsection.

The notion of non-Archimedean norm on a vector space is defined in [Reference Usher and Zhang48, Definition 2.2]. It is easy to check that $\mathcal {A}$ in (2.2) is a non-Archimedean norm on the k-vector space $CF(L_1,L_2)$ . Moreover, $CF(L_1,L_2)$ is orthogonal with respect to $\mathcal {A}$ .

Now, we are ready to apply [Reference Usher and Zhang48, Theorem 3.4] for the differential

$$\begin{align*}\delta: CF(L_1,L_2) \to CF(L_1,L_2).\end{align*}$$

Since $\delta $ is a linear self-mapping of an orthogonal vector space $CF(L_1,L_2)$ , one obtains a basis $\Sigma = \{\alpha _i, \beta _j, \gamma _j\}$ of $CF(L_1,L_2)$ satisfying

1. 1. $\partial \alpha _i = 0$ ,

2. 2. $\partial \gamma _j = \beta _j$ , and

3. 3. $\mathcal {A}(\gamma _1) - \mathcal {A}(\beta _1) \leq \mathcal {A}(\gamma _2) - \mathcal {A}(\beta _2) \leq \dots $ .

By using the above, we define the followings.

By Definition 7.1, Lemma 7.3 is obvious.

Lemma 7.3. Let $L_1$ and $L_2$ be a transversal pair of Lagrangians. Then, for any $\epsilon \geq 0$ ,

$$\begin{align*}b_\epsilon(L_1,L_2) \leq b_0(L_1, L_2) \leq |L_1 \cap L_2|.\end{align*}$$

It is well-known that $b_\epsilon $ is insensitive to small perturbations of the Lagrangians with respect to the Hofer distance. More precisely, Lemma 7.4 holds.

Lemma 7.4. Let $L_1'$ be a Lagrangian satisfying

• • $L_1'$ and $L_1$ are Hamiltonian isotopic to each other,

• • $d_H(L_1, L_1') < \frac {\delta }{2}$ with $\delta < \epsilon $ , and

• • $L_1'$ and $L_2$ are transversal to each other.

Then,

$$\begin{align*}b_{\epsilon+\delta}(L_1',L_2) \leq b_\epsilon(L_1,L_2) \leq b_{\epsilon - \delta}(L_1',L_2).\end{align*}$$

Now, we extend the barcode counting function $b_\epsilon (L_1,L_2)$ to a good pair $(L_1,L_2)$ defined in Definition 3.1. For a good pair, we set

$$\begin{align*}b_\epsilon(L_1, L_2) := \liminf_{d_H(L_2, L_2') \to 0} b_\epsilon(L_1, L_2'),\end{align*}$$

where the limit is taken over Lagrangians $L_2'$ such that $L_2' \pitchfork L_1$ . We also note that $L_2$ and $L_2'$ should be Hamiltonian isotopic so that the Hofer distance between them is defined.

7.2 Barcode entropy

In the rest of this paper, we consider the same situation as what we considered in Section 4. For the reader’s convenience, we review the setting.

Let $(\hat {W}, d\lambda )$ be a Weinstein manifold and let $\phi : \hat {W} \to \hat {W}$ be a compactly supported exact symplectic automorphism. Then, there is a Weinstein domain W such that

• • $\hat {W} = W \cup \left ( [1,\infty ) \times \partial W\right )$ ,

• • $\lambda |_{[1,\infty \times \partial W)} = r \alpha $ where r is a coordinate for $[1.\infty )$ and $\alpha := \lambda |_{\partial W}$ , and

• • the support of $\phi $ is contained in $\mathrm {Int}(W)$ .

Let $(L_1,L_2)$ be a good pair of Lagrangians with respect to W, that is,

$$\begin{align*}L_1 \cap L_2 \cap \left([1,\infty) \times \partial W \right) = \varnothing.\end{align*}$$

Since $\phi $ is the identity outside of W, $\left (L_1,\phi ^n(L_2)\right )$ is also a good pair for any $n \in \mathbb {Z}$ . Then, for a fixed $\epsilon $ , $b_\epsilon \left (L_1, \phi ^n(L_2)\right )$ is well-defined. We would like to define the barcode entropy of $\phi $ as the exponential growth rate of $b_\epsilon \left (L_1, \phi ^n(L_2)\right )$ as $n \to \infty $ .

To be more precise, let $\log ^+: \mathbb {Z}_{\geq 0} \to \mathbb {R}$ be the function defined as

$$\begin{align*}\log^+(k) = \begin{cases} 0 \text{ if } k =0, \\ \log(k) \text{ other wise, } \end{cases}\end{align*}$$

where the logarithm is taken base $2$ .

We note that Definition 7.5 is the same as the notion of relative barcode entropy in [Reference Cineli, Ginzburg and Gürel13], except a minor adjustment to our set up.

7.3 Barcode vs topological entropy

In this subsection, we prove that for any good pair $(L_1, L_2)$ , the barcode entropy of $\phi $ bounds the topological entropy of $\phi $ from below. The proof of Proposition 7.6 is almost same as the [Reference Cineli, Ginzburg and Gürel13, Proof of Theorem A].

Proposition 7.6 (= The second inequality in Proposition 1.4).

For any good pair $(L_1,L_2)$ ,

$$\begin{align*}h_{bar}(\phi;L_1,L_2) \leq h_{top}(\phi).\end{align*}$$

Proof. If $h_{bar}(\phi ;L_1,L_2) =0$ , then there is nothing to prove. Thus, let assume that $h_{bar}(\phi ;L_1,L_2)>0$ . We would like to show that if $\alpha \leq h_{bar}(\phi ;L_1,L_2)$ , then $\alpha \leq h_{top}(\phi )$ .

Let $\delta $ be a positive number. Since

$$\begin{align*}h_{bar}(\phi;L_1,L_2) := \lim_{\epsilon \searrow 0} h_\epsilon(\phi;L_1,L_2) \geq \alpha,\end{align*}$$

there is $\epsilon _0>0$ such that if $\epsilon < \epsilon _0$ , then $h_\epsilon (\phi ;L_1,L_2)> \alpha -\delta $ . We fix a positive number $\epsilon $ such that $2\epsilon < \epsilon _0$ .

Since

$$\begin{align*}h_{2 \epsilon}(\phi;L_1,L_2) = \limsup_{n \to \infty}\frac{1}{n} \log^+ b_{2 \epsilon}\left(L_1,\phi^n(L_2)\right)> \alpha - \delta,\end{align*}$$

there is an increasing sequence of natural numbers $\{n_i\}_{i \in \mathbb {N}}$ such that

(7.1) $$ \begin{align} b_{2\epsilon}\left(L_1,\phi^{n_i}(L_2)\right)> 2^{(\alpha-\delta)n_i}. \end{align} $$

Now, we apply Lemma 3.2 to the good pair $\left (L_1, \phi ^{n_i}(L_2)\right )$ . Then, one obtains a family of Lagrangians $\{L^s\}_{s \in B^d_{\epsilon , n_i}}$ such that

1. (i) $L_1$ and $L^s$ are Hamiltonian isotopic for all $s \in B^d_{\epsilon , n_i}$ ,

2. (ii) $d_H(L_1,L^s) < \frac {\epsilon }{2}$ for all $s \in B^d_{\epsilon , n_i}$ , and

3. (iii) $L^s \pitchfork \phi ^{n_i}(L_2)$ for almost all $s \in B^d_{\epsilon , n_i}$ .

We point out that by the argument in the proof of Theorem 4.1, we can choose a family satisfying (i)–(iii) for all $n_i$ . Let $\{L^s\}_{s \in B^d_\epsilon }$ denote a fixed Lagrangian tomograph.

Let

$$\begin{align*}N_i(s):= |L^s \cap \phi^{n_i}(L_2)|.\end{align*}$$

Then, one has

$$\begin{align*}\int_{B^d_\epsilon} N_i(s) ds \leq C \cdot \mathrm{Vol}(\phi^{n_i}(L_2) \cap W),\end{align*}$$

by applying Lemma 3.4. We note that C is a constant independent from $n_i$ .

From Lemmas 7.3 and 7.4 and Equation (7.1), we have

$$\begin{align*}2^{(\alpha-\delta)n_i} \leq b_{2\epsilon}\left(L_1,\phi^{n_i}(L_2)\right) \leq b_\epsilon\left(L^s,\phi^{n_i}(L_2)\right) \leq |L^s \cap \phi^{n_i}(L_2)| = N_i(s).\end{align*}$$

Then, by taking integration over $B^d_{\epsilon }$ , one has

$$\begin{align*}\mathrm{Vol}(B^d_\epsilon) \cdot 2^{(\alpha-\delta)n_i} \leq C \cdot \mathrm{Vol}\left(\phi^{n_i}(L_2) \cap W\right).\end{align*}$$

Since $\mathrm {Vol}(B^d_\epsilon )$ and C do not depend on $n_i$ ,

(7.2) $$ \begin{align} \alpha - \delta \leq \limsup_{i\to \infty} \frac{1}{n_i} \log^+ \mathrm{Vol}\left(\phi^{n_i}(L_2) \cap W \right) \leq h_{top}(\phi|_W). \end{align} $$

The last inequality holds due to Theorem 2.5, and because of the fact that

$$\begin{align*}\phi^{n_i}(L_2) \cap W = \phi^{n_i}(L_2 \cap W) = (\phi|_W)^{n_i}(L_2 \cap W).\end{align*}$$

Finally, we note that $\phi $ is compactly supported, and that $\mathrm {supp}(\phi ) \subset W$ . Thus,

$$\begin{align*}h_{top}(\phi) = h_{top}(\phi|_W).\end{align*}$$

Thus, one has

$$\begin{align*}\alpha - \delta \leq h_{top}(\phi).\end{align*}$$

This completes the proof.

7.4 Barcode vs categorical entropy

In the previous section, for an arbitrary good pair $(L_1,L_2)$ , we compared the barcode entropy of a triple $(\phi ;L_1,L_2)$ and the topological entropy of $\phi $ . As the result, we proved Proposition 7.6. In this subsection, we compare barcode and categorical entropy. However, in order to compare them, we should choose some specific pairs of Lagrangians.

First, we choose a stop $\Lambda $ giving a fully stopped partially wrapped Fukaya category $\mathcal {W}(W, \Lambda )$ . Let G be an embedded Lagrangian generating $\mathcal {W}(W,\Lambda )$ . As we did in the proof of Theorem 4.1, let $\varphi _0$ be a Hamiltonian isotopy satisfying

1. (A) $\left (\varphi _0(G),\phi ^n(G)\right )$ is a good pair for all $n \in \mathbb {N}$ ,

2. (B) $HW_\Lambda \left (G,\phi ^n(G)\right ) = HF\left (\varphi _0(G),\phi ^n(G)\right )$ for all $n \in \mathbb {N}$ .

As we showed before, one has

$$\begin{align*}h_{cat}(\Phi) = h_{cat}(\Phi_\Lambda) = \limsup_{n\to\infty} \frac{1}{n} \log^+ \dim HF\left(\varphi_0(G),\phi^n(G)\right).\end{align*}$$

Since $\dim HF\left (\varphi _0(G),\phi ^n(G)\right )$ equals the number of bars having infinite length, one has

$$\begin{align*}\dim HF\left(\varphi_0(G),\phi^n(G)\right) \leq b_\epsilon\left(\varphi_0(G),\phi^n(G)\right).\end{align*}$$

This induces Proposition 7.7.

Proposition 7.7 (= The first inequality in Proposition 1.4).

For G and $\varphi _0$ given above,

$$\begin{align*}h_{cat}(\Phi) \leq h_{bar}\left(\phi; \varphi_0(G), G\right).\end{align*}$$

7.5 Further questions

In this subsection, we discuss the questions given in Section 1.3 in more detail.

We also recall that

$$ \begin{align*} h_{top}(\phi) \geq \text{ the exponential growth rate of } \mathrm{Vol}\left(\phi^n(Y)\right), \end{align*} $$

for any compact submanifold Y by [Reference Newhouse40], [Reference Przytycki43]. And it is known by [Reference Yomdin50] that

(7.3) $$ \begin{align} h_{top}(\phi) = \sup_{\text{compact submanifold } Y \subset W} \left(\text{the exponential growth rate of } \mathrm{Vol}\left(\phi^n(Y)\right)\right). \end{align} $$

As one can see in the proof of Proposition 7.6, $h_{bar}(\phi ;L_1, L_2)$ bounds the exponential volume growth rate of $\phi ^n(L_2)$ from below. Thus,

$$\begin{align*}h_{top}(\phi) \geq h_{bar}(\phi;L_1, L_2).\end{align*}$$

As a generalization of Equation (7.3), one can ask whether the following equality holds or not:

$$\begin{align*}h_{top}(\phi) = \sup_{(L_1, L_2) \text{ is a good pair}} h_{bar}(\phi;L_1,L_2).\end{align*}$$

The supremum in the above equation runs over the set of all good pairs. As mentioned in Remark 7.8, it is easy to find a good pair $(L_1,L_2)$ such that $h_{bar}(\phi ;L_1,L_2)=0$ . Thus, we would like to remove such good pairs from the set where the supremum runs over, for computational convenience.

Finally, we ask whether the following equality holds or not:

$$\begin{align*}h_{top}(\phi) = \sup_{G, \varphi_0} h_{bar}\left(\phi;\varphi_0(G),G)\right),\end{align*}$$

where G is a generating Lagrangian and $\varphi _0$ is a Hamiltonian isotopy satisfying the conditions in Section 7.4.

On the other hand, one can ask a similar question for $h_{cat}(\phi )$ . More precisely, we ask whether the following equality holds or not:

$$\begin{align*}h_{cat}(\phi) = \inf_{G, \varphi_0} h_{bar}\left(\phi;\varphi_0(G),G)\right).\end{align*}$$