On the barcode entropy of Lagrangian submanifolds

To formulate the first theorem, we recall that a hyperbolic invariant set $K$ of a diffeomorphism $\psi$ is locally maximal if there exists a compact neighbourhood $U$ for which $K=\bigcap_{k\in\mathbbm{Z}}\psi^{k}({U})$; such a neighbourhood is called isolating. For every point $q$ in $K$ and $\delta>0$, we have local stable and unstable manifolds of size $\delta$, $W^{s}_{\delta}(q)$ and $W^{u}_{\delta}(q)$, see §3.2, which, for small enough $\delta$, are small Lagrangian disks passing through $q$. The following might be considered as a relative version of inequality (3), that is, of Theorem B in [13].

Note that every locally maximal compact hyperbolic set contains a topologically transitive one with the same topological entropy. Note also that the condition to contain $W_{\delta}^{u}(q)$ or $W_{\delta}^{s}(p)$ is a local condition. In fact, given $K$ as above, it is possible to modify any Lagrangian $L$ to one for which this condition is satisfied, keeping all but a small set inside $L$ fixed.

Together with [13, Thm. A], cf. (3), and a well-known result by Katok about the approximation of topological entropy by that of horseshoes in dimension 2, [27], Theorem 1 provides the following result; it resembles the Yomdin-Newhouse identity (in dimension $2$).

To motivate the next result, let me recall the following fact for surface diffeomorphisms with positive topological entropy. In general, for any continuous map $f:M\to M$ on a compact surface $M$ and any set of periodic orbits $\mathcal{O}\subset M$ of $f$, we obtain an induced action on the fundamental group in the complement $f_{\#}:\pi_{1}(M\setminus\mathcal{O})\to\pi_{1}(M,\setminus\mathcal{O})$ (up to some choices). Furthermore, given a homotopy class $\alpha$ of closed curves in $M\setminus\mathcal{O}$, we can consider the exponential growth rate $\Gamma_{\mathcal{O},\alpha}$ of $f^{n}_{\#}(\alpha)$ in $\pi_{1}(M\setminus\mathcal{O})$, in terms of the minimal length of words of generators necessary to represent $f^{n}_{\#}(\alpha)$, see [28, §3.1.c]. This growth rate is a lower bound, provided that $f$ is Lipschitz, to the growth rate of the length of closed curves representing $\alpha$. That is, for any closed curve $L$ with $[L]=\alpha$ we have that

Let $f$ be now a $C^{1+\alpha}$-diffeomorphism with $h_{\mathrm{top}}(f)>0$. By a result of Franks and Handel, [22], there exist a periodic orbit $\mathcal{O}$ and a homotopy class $\alpha$ of curves in the complement such that $\Gamma_{\mathcal{O},\alpha}(f)>0$. This result was improved by Alves and the author, [8]: For every $0<e<h_{\mathrm{top}}(f)$, there exist $\mathcal{O}$ and $\alpha$ as above such that

In particular, for any closed curve $L$ with $[L]=\alpha$,

The following result states that if we allow $\mathcal{O}$ to be a finite union of periodic orbits, then we can replace in that result the exponential length growth rate by relative barcode entropy.

Two fundamental and remarkable norms defined on groups of Hamiltonian diffeomorphisms are the Hofer-norm $\|\cdot\|_{\mathrm{Hofer}}$ and the spectral- (or $\gamma$-)norm $\gamma(\cdot)$. The former was first introduced by Hofer, [26], whereas the latter was introduced by Viterbo, [50], by means of generating functions, and by Schwarz, see [46], and Oh, see [41], using Floer theory. The Hofer-norm bounds the $\gamma$-norm from above, whereas the $C^{k}$ norms are not bounded by those norms in general. We refer to §2.4 and to the above references for definitions of these norms. They naturally induce distances on the groups of Hamiltonian diffeomorphisms, and furthermore also on any given Hamiltonian isotopy class $\mathcal{L}$ of Lagrangian submanifolds: the Hofer-distance and (exterior) $\gamma$-distance on $\mathcal{L}$ is defined as

see also [29] and references therein.

A remarkable property of the relative barcode entropy is that $\hbar(\psi;L_{0},L_{1})$ is lower semi-continuous with respect to $d_{\mathrm{Hofer}}$ and $d_{\gamma}$ in the pair $(L_{0},L_{1})$ varying in their Hamiltonian isotopy class. Together with these properties and estimate (3), Theorem 1 has the following consequence.

The next statement is a corollary of Theorem 3. For that, the reader should keep in mind that a small $d_{\gamma}$-perturbation of a closed curve in a surface does not necessarily preserve its homotopy class in a complement of a fixed finite set of points.

Finally, let me discuss some situations in which positive relative barcode entropy persists on a larger scale. We note that the positivity of barcode entropy for the situations considered so far had its origin in horseshoes, hence in some rather local dynamical features, and the property “$\hbar(\psi;L_{0},L_{1})>0$” is not invariant under Hamiltonian isotopies of the Lagrangian submanifolds in general. Also note, that while $\hbar(\psi;L_{0},L_{1})$ is lower semi-continuous with respect to the Hofer- or the $\gamma$-distance in the pair $(L_{0},L_{1})$, questions about stability properties in the diffeomorphism $\psi$ are more subtle. (One result in that direction, but which is obtained rather indirectly, is formulated in Remark 1.1.) In order to detect situations with stronger stability properties, it is convenient to consider a variant of relative barcode entropy. For $R>0$, let us define the $R$-strong relative barcode entropy of $(\psi,L_{0},L_{1})$ as

The following properties are either obvious or consequences of stability properties of barcodes with respect to $d_{\gamma}$, [29], see §2.4; strict monotonicity in $R$ (point (iii)) was pointed out to me by Çineli.

Hence, given a Hamiltonian isotopy class of Lagrangian submanifolds $\mathcal{L}$, the strong barcode entropies $\mathsf{H}^{R}(\psi;\mathcal{L}):=\mathsf{H}^{R}(\psi;L,L)$, $L\subset\mathcal{L}$, are well defined, and the relative barcode entropy is positive independent of the pair of Lagrangians in $\mathcal{L}$ if $\mathsf{H}^{R}(\psi;\mathcal{L})>0$. A necessary condition for the latter is that $\mathrm{Ham}(M,\omega)$ has infinite $d_{\gamma}$-diameter, which for example does not hold for the standard $\mathbbm{CP}^{n}$, [19]. To exhibit first examples with $\mathsf{H}^{R}(\psi;\mathcal{L})>0$, we consider the twisted products $(M,\omega)=(\Sigma\times\Sigma,\sigma\oplus-\sigma)$, where $(\Sigma,\sigma)$ is a closed symplectic surface with genus at least $2$. That the $d_{\gamma}$-diameter of $\mathrm{Ham}(M,\omega)$ is infinite follows from [47] and [29].³³3More generally, the $d_{\gamma}$-diameter of $\mathrm{Ham}(M,\omega)$ is infinite whenever $(\Sigma,\sigma)$ is a symplectic aspherical manifold, as was recently established by Mailhot in [32], and that result provided some motivation for the choice of examples here.

A pair of involutions as in the theorem exist for any closed symplectic surface considered above. Note also that with Proposition 1.5 (v), by passing to iterates of $\psi$, one finds, for all positive $R$ and $E$, examples where (5) holds. As a direct consequence of Proposition 1.5 (iv) and of [13, Thm. A]⁴⁴4In our situation, we will know more about the “homotopy classes” of intersections that generate barcode entropy. As a consequence, lower bounds on $h_{\mathrm{top}}$ can be obtained in a more elementary way, e.g. without the use of Yomdin’s theorem, cf. [2]. we obtain.

It was previously shown by Chor and the author, [17], that for a closed symplectic surface $(\Sigma,\sigma)$ of genus at least $2$, the assertion of Corollary 7 (for the Hofer-distance) holds for $\mathrm{Ham}(\Sigma,\sigma)$, using rather different methods. Interestingly enough, and to the best of my knowledge, the following question remains open.

Idea of proofs. As Theorem 1 is a relative analogue of Theorem B of [13], also the proof is very much a relative analogue of the proof in [13]. In a nutshell, Theorem 1 can be established by an interplay of arguments from hyperbolic dynamics together with $C^{1}$-bounds for pseudo-holomorphic curves with Lagrangian boundary conditions: by results going back to Bowen, the number of orbit segments from local unstable to local stable manifolds in a locally maximal, topological transitive hyperbolic set $K$ is bounded from below by its topological entropy. Moreover, a combination of $C^{1}$-estimates for Floer strips and the shadowing theorem shows that the energy that is needed to “connect” two chords in $K$ or to “leave” an isolating neighbourhood is uniformly bounded from below. Together, the estimates of Theorem 1 will be obtained.

The proof of Theorem 3 follows a similar strategy with two additional inputs. First, we apply the fact, recently proved in [36], that there exist a specific collection of horseshoe orbits $\mathcal{O}$ and a pair of curves in the complement such that the growth of chords of the suspension flow connecting those curves and which are unique in their homotopy class in the complement of the link $\mathcal{L}_{\mathcal{O}}$ induced by $\mathcal{O}$, is exponential; moreover, all these chords can be encoded with the help of specific Markov rectangles. And second, we use that the Lagrangian Floer homology can be filtered in homotopy classes of chords in the complement of $\mathcal{L}_{\mathcal{O}}$. Together with the estimates already used in the proof of Theorem 1, a “crossing energy lemma” is proved for certain Floer strips crossing $\mathcal{L}_{\mathcal{O}}$, where we have flexibility in the choice of the pair of Lagrangian submanifolds while keeping the diffeomorphism fixed.

Finally, Theorem 6 is obtained by considering a diffeomorphism $\psi$ supported in the neighbourhood of the unit sphere bundle of $\Sigma$ embedded in a neighbourhood of the diagonal in $M=\Sigma\times\Sigma$. The support of $\psi$ is foliated by leaves diffeomorphic to that sphere bundle and $\psi$ restricted to a leaf will coincide with the time $T$ map of the geodesic flow for a hyperbolic metric, where $T$ varies between leaves, monotonically increasing first, and decreasing later. The non-trivial chords relative $L$ will come in pairs with action difference growing linearly with iterations of $\psi$. The special topological situation then prevents the existence of Floer strips connecting chords between different pairs, as long as one chord belongs to a special subset of pairs. The number of pairs in that subset grows exponentially, which establishes the result.

Plan of the paper. We start in Section 2 with some preliminaries on Lagrangian Floer homology and relative barcode entropy. Section 3 contains a lemma about Floer strips with small energy and the proof of Theorem 1. In Section 4, special properties of Lagrangian Floer homology and horseshoes in dimension 2 are discussed, before a proof of Theorem 3 is given. In Section 5, we then construct the examples that establish Theorem 6.

Acknowledgements. The author is grateful to Marcelo Alves, Joé Brendel, Erman Çineli, Viktor Ginzburg, Başak Gürel, and Leonid Polterovich for useful discussions. Special thanks go to Urs Fuchs for helpful comments concerning properties of Floer strips with small energy.

Let $(M,\omega)$ be a symplectic manifold. To keep the exposition simple, we assume that $M$ is closed, but as in [13], the discussion extends to open $M$ if some assumptions are made on the ends of $(M,\omega)$ and if we consider compactly supported Hamiltonians. Let $\psi:M\to M$ be a Hamiltonian diffeomorphism. This means that $\psi=\phi_{H}^{1}$ is the time-$1$ map of a (time-dependent) flow $\phi_{H}^{t}$ of a $1$-periodic Hamiltonian $H:\mathbbm{R}/\mathbbm{Z}\times M$, i.e. $\frac{d}{dt}\phi^{t}_{H}(x)=X_{H_{t}}(\phi^{t}_{H}(x))$ ($(t,x)\in\mathbbm{R}\times M)$, where $\omega(X_{H_{t}}(x),\cdot)=-dH_{t}(x)$, and $H_{t}(x):=H(x,t)$. Let $Y:=S^{1}\times M$, where we identify $S^{1}\cong\mathbbm{R}/\mathbbm{Z}$. Fix once and for all a Hamiltonian function $H$ such that $\phi_{H}^{1}=\psi$. This defines a lift to the suspension flow $\varphi:\mathbbm{R}\times Y\to Y$, defined by

the flow of the vector field $X=\partial_{t}\oplus X_{H_{t}}$ on $Y$. For an orbit segment $\eta:[0,k]\to M$, $n\in\mathbbm{N}$, of the Hamiltonian flow $\phi_{H}^{t}$, we obtain an orbit segment $\widehat{\eta}:[0,k]\to Y$, $\widehat{\eta}(t):=(t,\eta(t))$ of $\varphi$.

If not explicitly stated otherwise, we will consider a fixed metric $g$ on $M$ and denote by $d$ resp. $\overline{d}$ the distance function on $M$ resp. $Y=S^{1}\times M$ induced by $g$ resp. the product metric $g_{\mathrm{Eucl}}\times g$.

We briefly recall relevant notions for Lagrangian Floer homology. The construction of Lagrangian Floer homology goes back to Floer [21] and in the monotone setting to Oh [40], the setting that is relevant here. More specifically, we closely follow setup and notation of [47] and [13]. In the following, all Lagrangian submanifolds $L$ will be closed and monotone. This means that there exists $\kappa>0$ such that $\langle\omega,A\rangle=\kappa\langle\mu_{L},A\rangle$ for all $A\in\pi_{2}(M,L)$, where $\mu_{L}$ is the Maslov class of $L$. We also require that $N_{L}\geq 2$, where the minimal Chern number $N_{L}$ is defined as the positive generator of the subgroup in $\mathbbm{Z}$ generated by the $\langle\mu_{L},A\rangle$ for all $A\in\pi_{2}(M,L)$ that can be represented by cylinders with boundary in $L$, and $N_{L}=\infty$ if this subgroup is trivial, see also [47, 13]. We also denote by $\mathfrak{h}(M,L)=\kappa N_{L}\in(0,+\infty]$ the minimal area of a non-trivial cylinder with boundary in $L$.

Let $L_{0}\subset M$ be a Lagrangian submanifold as above, and let $L_{1}=\phi^{1}_{G}(L_{0})$ be Hamiltonian isotopic to $L_{0}$, for some Hamiltonian $G$ that we fix from now on. Assume that the pair $(L_{0},L_{1})$ is non-degenerate; we recall that this means that $L_{0}$ and $\psi^{-k}(L_{1})$ intersect transversally, for all $k\in\mathbbm{N}$. For given $k\in\mathbbm{N}$, the construction of the Floer chain complexes $\mathcal{C}(L_{0},L_{1},k)=\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))$ can be sketched as follows. Let $\mathcal{P}^{k}=\mathcal{P}(L_{0},\psi^{-k}(L_{1}))$ be the space of smooth paths $x:[0,1]\to M$ with $x(0)\in L_{0}$, $x(1)\in\psi^{-k}(L_{1})$. Fix a system $\gamma_{\mathfrak{c}}\in\mathcal{P}^{k}$ of representatives of the homotopy classes $\mathfrak{c}\in\pi_{1}(M;L_{0},\psi^{-k}(L_{1}))$. Denote by $\widetilde{\mathcal{P}}^{k}$ the set of pairs $(x,[v_{x}])$, where $x\in\mathcal{P}^{k}$ and $[v_{x}]$ is a capping up to a certain equivalence relation. Here a capping $v_{x}$ is a $C^{1}$-map $v_{x}:[0,1]^{2}\to M$ with $v(\cdot,0)\in L_{0}$, $v(\cdot,1)\in\psi^{-k}(L_{1})$ and $v(0,t)=\gamma_{\mathfrak{c}}$ and $v(1,t)=x(t)$. Two cappings are equivalent if their $\omega$-area and Maslov index coincide; the Maslov index is well-defined given a choice of symplectic trivializations of $\gamma_{\mathfrak{c}}^{*}TM$, see [47]. Given $\hat{x}=(x,[u_{x}])\in\widetilde{\mathcal{P}^{k}}$, we can glue annuli $v:S^{1}\times[0,1]\to M$, with $v(s,0)\in L_{0}$, $v(s,1)\in\psi^{-k}(L_{1})$, to $v_{x}$ and obtain recappings $v\#\hat{x}=(x,[v\#v_{x}])$. We consider $\mathcal{A}:\widetilde{\mathcal{P}}^{k}\to\mathbbm{R}$, defined by

The critical points of $\mathcal{A}$ are those $(x,[v_{x}])$ for which $x\in L_{0}\cap\psi^{-k}(L_{1})$ is constant. We define $\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))$ as in [13]: fix for each $x\in L_{0}\cap\psi^{-k}(L_{1})$ one equivalence class of cappings $[v_{x}]$ of $x$. Denote the set of those pairs by $\mathcal{X}^{k}\subset\widetilde{\mathcal{P}}^{k}$, and let $\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))$ be the vector space generated by the elements in $\mathcal{X}^{k}$ over the universal Novikov field $\Lambda$, which is formed by the formal sums $\lambda=\sum_{j\geq 0}f_{j}T^{a_{j}}$, $f_{j}\in\mathbbm{Z}_{2}$, $a_{j}\in\mathbbm{R}$, subject to the condition $\#\{j\,|\,f_{j}\neq 0,a_{j}\leq C\}<\infty$ for all $C\in\mathbbm{R}$. To define the boundary map $\partial:\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))\to\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))$, one considers, for a choice of $S^{1}$-family $J=(J_{t})_{t\in S^{1}}$ of almost complex structures compatible with $\omega$ and critical points $\hat{x}_{\pm}=(x_{\pm},[v_{{\pm}}])$ of $\mathcal{A}$, the moduli space $\mathcal{M}_{J}(\hat{x}_{-},\hat{x}_{+},L_{0},\psi^{-k}(L_{1}))$ of holomorphic strips $u:\mathbbm{R}\times[0,1]\to M,$

subject to boundary conditions $u(s,0)\subset L_{0}$, $u(s,1)\in\psi^{-k}(L_{1})$ ($s\in\mathbbm{R}$), asymptotics $\lim_{s\to\pm\infty}u(s,t)\equiv x_{\pm}$, and for which the glued map $v_{-}\#u$ is equivalent to $v_{+}$. If $J$ is such that the moduli spaces are transversally cut out, which holds generically, one can show that if the Maslov index difference of $\hat{x}$ and $v\#\hat{x}_{j}$ is $1$, $\mathcal{M}_{J}(\hat{x}_{-},\hat{x}_{+},L_{0},\psi^{-k}(L_{1}))/\mathbbm{R}$ is finite, where we are modding out by the canonical $\mathbbm{R}$-action on solutions of (7). This makes it possible to define the boundary map $\partial$, defined on the generators $\mathcal{X}^{k}$ as $\partial\hat{x}=\sum\lambda_{j}\hat{x}_{j}$, where $\lambda_{j}=\sum_{v}f_{v}T^{\omega(v)}\in\Lambda$. Here the sum is taken over all recappings $v$ of $\hat{x}_{j}$ such that the Maslov index difference of $\hat{x}$ and $v\#\hat{x}_{j}$ is $1$, and $f_{v}$ is the number of elements (mod $2$) in $\mathcal{M}_{J}(\hat{x},v\#\hat{x}_{j},L_{0},\psi^{-k}(L_{1}))/\mathbbm{R}$. One shows that $\partial$ is indeed a boundary map $(\partial^{2}=0)$ and that the homology denoted by $\mathrm{HF}(L_{0},\psi^{-k}(L_{1}))$ does not depend on the choice of $J$.

The homology is invariant under Hamiltonian isotopy: a Hamiltonian isotopy $L^{s}_{0}=\phi_{F}^{s}(L_{0})$, $F:M\times\mathbbm{R}/\mathbbm{Z}\to\mathbbm{R}$, $s\in[0,1]$, from $L_{0}$ to $L^{\prime}_{0}:=L^{1}_{0}$ with $(L^{\prime}_{0},L_{1})$ non-degenerate, induces for every $k\in\mathbbm{N}$ a chain map $f_{0}:\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))\to\mathrm{CF}(L^{\prime}_{0},\psi^{-k}(L_{1}))$, defined by counting holomorphic strips $u:\mathbbm{R}\times S^{1}\to M$ (with $s$-dependent $J$) in a similar manner as above, now with moving Lagrangian boundary conditions $u(s,0)\in L^{\beta(s)}_{0}$ and $u(s,1)\in\psi^{-k}(L_{1})$ $(s\in\mathbbm{R})$, where $\beta:\mathbbm{R}\to[0,1]$ is a smooth non-decreasing surjective function, constant outside a compact interval [40, §5], see also [10, §3.2]. Similarly, one defines $g_{0}:\mathrm{CF}(L^{\prime}_{0},\psi^{-k}(L_{1}))\to\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))$ and one obtains that $g_{0}\circ f_{0}$ is chain homotopic to the identity. Analogously, for a Hamiltonian isotopy $L_{1}^{s}$, $s\in[0,1]$, from $L_{1}$ to $L^{\prime}_{1}$, one obtains for all $k\in\mathbbm{N}$ a chain map $f_{1}:\mathrm{CF}(L_{0},\psi^{-k}(L_{1}))\to\mathrm{CF}(L_{0},\psi^{-k}(L^{\prime}_{1}))$ with similar properties.

Given a holomorphic strip $u\in\mathcal{M}_{J}(\hat{x}_{-},\hat{x}_{+},L_{0},\psi^{-k}(L_{1}))$, we will consider the rescaled strip $\check{u}:\mathbbm{R}\times[0,k]\to M$ defined by $\check{u}(s,t)=u(\frac{1}{k}s,\frac{1}{k}t)$. The strip $\check{u}$ is holomorphic, that is, it satisfies (7) with $J_{t}$ replaced by $J_{\frac{1}{k}t}$. By composing with the Hamiltonian flow, we obtain

The strip $\hat{u}$ satisfies the Floer equation

where $\hat{J}_{t}:=(\phi_{H}^{t})_{*}J_{\frac{1}{k}t}$, $t\in[0,k]$. The asymptotics of $\hat{u}$ are chords from $L_{0}$ to $L_{1}$ ($\lim_{s\to\pm\infty}\hat{u}(s,t)=\phi^{t}_{H}(x_{\pm})$, $t\in[0,k]$) and $\hat{u}$ satisfies $\hat{u}(s,0)\in L_{0}$, $\hat{u}(s,k)\in L_{1}$, $s\in\mathbbm{R}$. This identification extends in a natural way to the situation of moving boundary conditions and $s$-dependent $J$.

Additionally, we consider also the maps

With respect to a suitable (time-independent) $\widetilde{J}$, the lift ${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{u}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{u}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{u}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{u}$}}}}$ is $\widetilde{J}$-holomorphic. Here, $\widetilde{J}$ is defined at a point $(s,t,p)\in\mathbbm{R}\times[0,k]\times M$ by $\widetilde{J}_{(s,t,p)}(\partial_{s})=\partial_{t}+X_{H}(s,t,p)$, where $X_{H}(s,t,p)$ is the (time-independent) horizontal lift of $X_{H_{t}}(p)$ to $\mathbbm{R}\times[0,k]\times M$, and by the condition that $\widetilde{J}_{(s,t,p)}(v)$ is the horizontal lift of $\hat{J}_{t}(v)$ to $\{s\}\times\{t\}\times M$. The boundary components of ${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{u}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{u}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{u}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{u}$}}}}$ then lie in $\mathbbm{R}\times\{0\}\times L_{0}$ and $\mathbbm{R}\times\{k\}\times L_{0}$.