1.1 Free homotopy classes of periodic orbits and forcing

Let $M$ be a closed oriented surface and $f:M \to M$ a homeomorphism isotopic to the identity. The topological entropy $h_{\mathrm {top}}(f)$ is one of the most widely used measures for the complexity of the dynamics of $f$ and can be defined as follows: for a fixed metric $d$ on $M$, denote by $N_{\mathrm {top}}(f,n,\epsilon )$ the maximal cardinality of a set $X$ in $M$ such that $\sup _{0\leq k\leq n} d(f^k(x), f^k(y)) \geq \epsilon$ for all $x,y \in X$ with $x\neq y$. The topological entropy of $f$ is then

\[ h_{\mathrm{top}}(f) := \lim_{\epsilon \to 0} \limsup_{n\to \infty} \frac{\log(N_{\mathrm{top}}(f,n,\epsilon))}{n}, \]

which is well defined and independent of the metric $d$. A related measure for dynamical complexity is the growth of periodic points: if $N_{\mathrm {per}}(f,n)$ denotes the number of periodic points of $f$ of period at most $n$, then the (exponential) growth rate of periodic points is defined as $\mathrm {Per}^{\infty }(f) := \limsup _{n \to \infty } ({\log (N_{\mathrm {per}}(f,n))}/{n})$.

Although it is in general very hard to calculate or estimate the topological entropy, remarkably, the existence of certain types of periodic points of $f$ implies lower bounds on $h_{\mathrm {top}}(f)$. The following result in dimension $1$ is a typical example of this forcing phenomenon: if a continuous mapping $g$ on $[0,1]$ admits a periodic point of period $n$ which is not a power of 2, then $h_{\mathrm {top}}(g)> ({1}/{n})\log (2)$ (see [Reference Bowen and FranksBF76]). Passing to dimension $2$, in order to deduce some interesting dynamical information of $f$, the period alone is not sufficient and one has to take into account further characteristics of a periodic point. It is natural to require that the desired bounds are invariant under any isotopy of $f$ relative to the periodic orbit, which amounts to considering the braiding information, or braid type, of the periodic orbit in the suspension flow. Thurston–Nielsen theory provides a natural framework for understanding the dynamics that is forced by a given braid type, namely, dynamically minimal maps for a braid type are given by the Thurston–Nielsen canonical form. There are algorithmic methods to obtain the latter, although it sometimes becomes difficult to apply them in practice, especially when passing to families of braid types. Entropy bounds for specific (families of) braid types have been investigated by many researchers (see, for example, [Reference HallHal94, Reference SongSon05, Reference Hironaka and KinHK06] and references therein). The complete braiding information of an orbit, in particular if one considers orbits of higher complexity, is somewhat hard to overlook, and one simplification is to project braids back to the surface, in other words, to look for braid-type invariants that can be presented purely in terms of the homotopy class of the curve that the orbit traces on $M$ through an identity isotopy of $f$. It is that kind of invariants that we will consider in this paper. Especially for dealing with bounds related to that kind of invariants, the recently developed forcing theory of Le Calvez and Tal [Reference Le Calvez and TalLeCT18] building upon Le Calvez’s theory of transverse foliations [Reference Le CalvezLeC05] is perfectly suited. Lower bounds for $h_{\mathrm {top}}(f)$ and $\mathrm {Per}^{\infty }(f)$ of that kind are contained in [Reference Le Calvez and TalLeCT18, Reference Le Calvez and TalLeCT22, Reference Da SilvaDaS19], as applications of their methods, and in [Reference DowdallDow11] using Thurston–Nielsen theory. Our results improve and extend part of these bounds, and we work closely within the theory in [Reference Le Calvez and TalLeCT18]. For related results see also the recent work [Reference Guihéneuf and MilitonGM22]. It contains a proof of the existence of topological horseshoes if the homotopy class induced by the orbit cannot be represented by a multiple of a simple curve, with additional properties of the horseshoe depending on further specifics of that class.

Let us introduce some essential notions. Let $I= (I_t)_{t\in [0,1]}$ be an identity isotopy for $f$ (i.e. a continuous path from the identity to $f$), and consider the set $\mathrm {dom}(I) = M\setminus \mathrm {Fix}(I)$, where $\mathrm {Fix}(I)=\{x \in M \, |\, I_t(x) \equiv x \text { for all } t\in [0,1]\} \subset \mathrm {Fix}(f)$ are the points that are fixed throughout the isotopy $I$. We denote by $\widehat {\pi }(X)$ the set of free homotopy classes of loops in a space $X$, and for any loop $\Gamma :S^{1} \to X$, by $[\Gamma ] = [\Gamma ]_{\widehat {\pi }(X)}$ its free homotopy class in $\widehat {\pi }(X)$. We say that $\alpha \in \widehat {\pi }(\mathrm {dom}(I))$ is primitive if there is no representative of $\alpha$ that multiply covers another loop. For $m\in \mathbb {N}$, denote by $m\alpha$ the free homotopy class that is represented by the $m$-fold iteration of a loop that represents $\alpha$. For any periodic point $x\in \mathrm {dom}(I)$ of $f$, say of period $q \in \mathbb {N}$, consider the loop $I^q(x)$ given by the concatenation of the paths $I_t(x)_{t\in [0,1]}$, $I_t(f(x))_{t\in [0,1]}, \dots$, and $I_t(f^{q-1}(x))_{t\in [0,1]}$. The loop $I^q(x)$ defines a free homotopy class $[I^q(x)]= \alpha \in \widehat {\pi }(\mathrm {dom}(I))$ of loops in $\mathrm {dom}(I)$. An identity isotopy $I$ is maximal if, for any fixed point $y\in \mathrm {dom}(I)$ of $f$, the loop $I(y)= I^1(y)$ is not contractible in $\mathrm {dom}(I)$ (cf. § 2.3). Let $\Gamma : S^1 \to \mathrm {dom}(I)$ be a loop and denote by $\mathcal {S}(\Gamma ):= \{y \in \mathrm {dom}(I)\, | \, y = \Gamma (t) = \Gamma (t'), t\neq t'\}$ the set of self-intersections of $\Gamma$, and by $\mathrm {si}(\Gamma ) = \#\mathcal {S}$ its cardinality. The geometric self-intersection number of $\alpha$ is defined as $\mathrm {si}_{\mathrm {dom}(I)}(\alpha ) := \min \mathrm {si}(\Gamma )$, where the minimum is taken over all smooth loops $\Gamma$ with $[\Gamma ] = \alpha$ in $\widehat {\pi }(\mathrm {dom}(I))$ that are in general position, that is, each self-intersection $y\in \mathcal {S}(\Gamma )$ is transverse and $\# \Gamma ^{-1}(y) = 2$.

Theorem 1.1 Let $M$ be a closed oriented surface, $f:M \to M$ a homeomorphism isotopic to the identity, and $I$ a maximal identity isotopy for $f$. Let $\alpha \in \widehat {\pi }(\mathrm {dom}(I))$ be primitive with $\mathrm {si}_{\mathrm {dom}(I)}(\alpha ) \neq 0$. If there is a $q$-periodic point $x$ of $f$ in $\mathrm {dom}(I)$ with $[I^q(x)] = m\alpha \in \widehat {\pi }(\mathrm {dom}(I))$ for some $m\in \mathbb {N}$, then both $\mathrm {Per}^{\infty }(f)$ and $h_{\mathrm {top}}(f)$ are at least equal to

\[ \frac{m}{q}\max\biggl\{\frac{\log(\mathrm{si}_{\mathrm{dom}(I)}(\alpha) + 1)}{16}, \frac{\log(2)}{2}\biggr\}. \]

If $M = S^2$ and $\mathrm {si}_{\mathrm {dom}(I)}(\alpha ) \neq 0$, a fixed positive lower bound for $h_{\mathrm {top}}(f)$ and $\mathrm {Per}^{\infty }(f)$ was obtained in [Reference Le Calvez and TalLeCT18, Theorem 41] and improved in [Reference Le Calvez and TalLeCT22, Reference Da SilvaDaS19]. Theorem 1.1 improves these lower bounds, the main addition here being that the bounds grow with the complexity of $\alpha$. We note that for many choices of $M$ and $\alpha$, similar lower bounds were obtained by Dowdall in [Reference DowdallDow11] using Thurston–Nielsen theory.

Theorem 1.1 implies similar lower bounds that are independent of the isotopy $I$. If $M$ has genus greater than or equal to $2$, then, since the space of homeomorphisms isotopic to the identity is contractible, the free homotopy class $\alpha := [I^q(x)]_{\widehat {\pi }(M)}$ in $\widehat {\pi }(M)$ for a $q$-periodic point $x \in M$ of $f$ does not depend on the choice of identity isotopy $I$, and we may say that $x$ is a $q$-periodic point of class $\alpha$.Footnote ² Theorem 1.1 and a result in [Reference Béguin, Crovisier and Le RouxBCLR20] about existence of maximal isotopies (cf. § 2.3) yield the following theorem (see § 4).

Theorem 1.2 Let $M$ be a closed oriented surface of genus $g\geq 2$ and $\alpha$ a primitive class in $\widehat {\pi }(M)$ with $\mathrm {si}_M(\alpha ) \neq 0$. Let $f:M \to M$ be a homeomorphism that is isotopic to the identity. If $f$ has a $q$-periodic point of class $\alpha$, then $\mathrm {Per}^{\infty }(f)$ and $h_{\mathrm {top}}(f)$ are at least equal to

\[ {1}/{q}\max\biggl\{\frac{\log(\mathrm{si}_M(\alpha) + 1)}{16}, \frac{\log(2)}{2}\biggr\}. \]

In other words, a periodic point of class $\alpha$ that admits geometric self-intersections forces lower bounds for topological entropy and exponential orbit growth, and the more self-intersections the higher the bounds.

Theorem 1.2 raises the following question. What can be said about the classes in $\widehat {\pi }(M)$ of those periodic points that are forced by the periodic point $x$ of class $\alpha$? We address this question in connection with growth and consider the exponential homotopical orbit growth rate given by

\[ \mathrm{H}^{\infty}(f):=\limsup_{n \to \infty} \frac{\log(N_{\mathrm{h}}(f,n))}{n}, \]

where $N_{\mathrm {h}}(f,n)$ is the number of distinct classes of periodic points of period less than $n$. Do $\mathrm {H}^{\infty }(f)$ and $\mathrm {Per}^{\infty }(f)$ satisfy similar bounds? While the proof of Theorem 1.2 does not give a positive lower bound on $\mathrm {H}^{\infty }(f)$ in terms of $\mathrm {si}_M(\alpha )$, we can bound $\mathrm {H}^{\infty }(f)$ in terms of another invariant of $\alpha$. We adapt a construction of Turaev [Reference TuraevTur78, Reference TuraevTur91] and then use it to define a growth rate $\mathrm {T}^{\infty }(\alpha )$. We explain this construction briefly here (see § 5 for details).

Denote by $\widehat {\pi }(M)^*$ the set of non-trivial classes in $\widehat {\pi }(M)$. A loop $\Gamma :[0,1] \to M$ in general position representing a class $\alpha \in \widehat {\pi }(M)^*$ splits at each self-intersection point $y \in \mathcal {S}(\Gamma )$ into two (oriented) closed loops $u^y_1$ and $u^y_2$ based at $y$, representing two classes $\alpha ^y_1, \alpha ^y_2 \in \widehat {\pi }(M)$, where we choose the labeling such that the initial tangent points of $u^y_1$ and $u^y_2$ define the orientation of $\Sigma$. Denoting by $\mathcal {Y}$ the intersection points for non-trivial $\alpha ^y_1$ and $\alpha ^y_2$,

\[ v(\alpha) = \sum_{y \in \mathcal{Y}} \alpha^y_1 \otimes \alpha^y_2 - \alpha^y_2 \otimes \alpha^y_1 \in \mathbb{Z}[\widehat{\pi}(M)^*] \otimes \mathbb{Z}[\widehat{\pi}(M)^*] \]

defines (by linear extension) Turaev's cobracket on the free $\mathbb {Z}$-module over $\widehat {\pi }(M)^*$. We consider a variant of this construction, and assume additionally that $x_0 = \Gamma (0) = \Gamma (1)$ is not an intersection point of $\Gamma$. For each self-intersection point $y= \Gamma (t) = \Gamma (t'), t< t'$, by composing with the paths $\Gamma _{[0,t]}$ and its inverse $\overline {\Gamma _{[0,t]}}$ we obtain loops $\Gamma _{[0,t]}u ^y_i \overline {\Gamma _{[0,t]}}$, $i=1,2$, based at $x_0 = \Gamma (0) = \Gamma (1)$. These loops define elements $a^y_i \in \pi _1(M,x_0)$, $i=1,2$ (see Figures 12 and 13).

Let $g \in \pi _1(M,x_0)$ be the element represented by $\Gamma$. Denote by $\pi _1(M,x_0)^*_g$ the set of $g$-equivalence classes of the non-trivial elements in $\pi _1(M,x_0)$, where we say that two elements are $g$-equivalent if one is a conjugation of the other by a multiple of $g$, and denote the equivalence class by $[\cdot ]_g$. The element

\[ \mu(g) = \sum_{y \in \mathcal{Y}} [a^y_1 ]_g \otimes [ a^y_2]_g - [a^y_2]_g \otimes [a^y_1]_g\in \mathbb{Z}[\pi_1(M,x_0)_g^*] \otimes \mathbb{Z}[\pi_1(M,x_0)_g^*] \]

is independent of the choice of the loop $\Gamma$ based at $x_0$ that represents $g$; moreover, $\mu (g)$ behaves well with respect to conjugation (see Lemma 5.1). One actually obtains via $\mu$ an invariant of $\alpha \in \widehat {\pi }(M)$ that is a (strict) refinement of the invariant $v$. We continue to define a growth rate as follows. First, for a subset $S=\{s_1, \dots, s_m\} \subset \pi _1(M,x_0)$, let $\widehat {N}(n,S)$ be the number of distinct conjugacy classes of elements in $\pi _1(M,x_0)$ that can be written as a product of at most $n$ factors from $S$, and define $\Gamma (S):= \limsup _{n\to \infty } ({\log (\widehat {N}(n,S))}/{n})$. Moreover, given $g\in \pi _1(M,x_0)$ and a set $\mathfrak {S}$ of $g$-equivalence classes of elements in $\pi _1(M,x_0)$, we define $\Gamma (\mathfrak {S},g) = \inf \, \Gamma (S)$, where the infimum is taken over all sets $S\subset \pi _1(M,x_0)$ with $\# S = \# \mathfrak {S}$ and such that each element in $S$ represents a different $g$-equivalence class in $\mathfrak {S}$. Now, writing $\mu (g) = \sum _{\mathfrak {a}, \mathfrak {b} \in \pi _1(M,x_0)^*_g} k_{\mathfrak {a}, \mathfrak {b}}\big ( \mathfrak {a} \otimes \mathfrak {b}\big )$, with $k_{\mathfrak {a},\mathfrak {b}} \in \mathbb {Z}$, we define the complexity $\mathrm {Comp}(\mu (g))$ as the collection of terms $\mathfrak {a} \otimes \mathfrak {b}$ with $k_{\mathfrak {a},\mathfrak {b}} >0$. Let $\mathrm {Comp}_+(\mu (g)) = \{ \mathfrak {a} \, | \, \exists \mathfrak {b}\, : \, \mathfrak {a} \otimes \mathfrak {b} \in \mathrm {Comp}(\mu (g))\}$ and $\mathrm {Comp}_+(\mu (g)) = \{\mathfrak {b} \, | \,\exists \mathfrak {a} \, : \, \mathfrak {a} \otimes \mathfrak {b} \in \mathrm {Comp}(\mu (g)) \}$, and define

\[ \Gamma^{g} := \min_{\pm} \min_{\mathfrak{S}}\big \{\Gamma(\mathfrak{S} \cup [g]_g, g)\, \big | \, \mathfrak{S} \subset \mathrm{Comp}_{\pm}(\mu(g)), \, \#\mathfrak{S}= \big \lceil \tfrac{1}{2}\#\mathrm{Comp}(\mu(g)) \big \rceil \big \}. \]

One shows that $\Gamma ^g$ is actually invariant under conjugation, and hence defines a growth rate $\mathrm {T}^{\infty }(\alpha ) \in [0,+\infty )$ associated to each free homotopy class $\alpha$ of loops in $M$.

We obtain the following result.

By a version of Ivanov's inequality we have that $h_{\mathrm {top}}(f) \geq \mathrm {H}^{\infty }(f)$ ([Reference IvanovIva82, Reference JiangJia96]; see also [Reference AlvesAlv16a]). Hence, Theorem 1.3 provides another lower bound on $h_{\mathrm {top}}(f)$ in terms of the complexity of $\alpha$. In § 7.3 we exhibit an infinite family of classes $\alpha$ in $\widehat {\pi }(M)$ such that $\mathrm {T}^{\infty }(\alpha ) \geq \log ({\mathrm {si}_M(\alpha )/4})$ (see Lemmas 7.8 and 7.9). In particular, the bound in Theorem 1.3 sometimes turns out to be significantly better than that in Theorem 1.2.

We outline the general strategy for the proofs of Theorems 1.1–1.3. Assume there is a $q$-periodic point $x$ of $f$ that is in a primitive class $\alpha$ with $\mathrm {si}_M(\alpha ) >0$. By the results in [Reference Béguin, Crovisier and Le RouxBCLR20] one can choose a maximal identity isotopy $I$ for $f$. Le Calvez’s theory of transverse foliations yields a singular foliation $\mathcal {F}$ on $M$ that is transverse to $I$ (see § 2). This means, in particular, that there is a loop $\Gamma$ transverse to $\mathcal {F}$ which is freely homotopic to $I^q(x)$ in $\mathrm {dom} (I)$, and we say that $\Gamma$ is associated to $x$. Le Calvez and Tal's results provide methods to manipulate $\Gamma$ in order to obtain other transverse loops associated to periodic points of $f$. Roughly speaking, in good situations such loops might be created by starting at $\Gamma (0)$, following $\Gamma$ positively transverse to $\mathcal {F}$, stopping at some intersection point $y$ which is reached for the first time, and finally continuing along $\Gamma$ in positive direction after a turn at $y$. The turn is to the left or to the right, depending on which direction is positively transverse to the foliation. In other words, one creates a shortcut of $\Gamma$ at $y$. Moreover, one might iterate this procedure and create shortcuts at several self-intersection points of $\Gamma$. The question whether the created loops are associated to periodic points is a bit subtle and one crucial assumption is that the intersections where the shortcuts are created are $\mathcal {F}$-transverse (see § 2 for the definition). Moreover, especially if several self-intersections are considered, it is important to understand what the maximal ‘length’ of a subpath of a lift of $\Gamma$ to the universal cover of $\mathrm {dom}(I)$ is such that this subpath is $\widetilde {\mathcal {F}}$-equivalent to a subpath of another lift of $\Gamma$ (see § 2 for the definitions). A better upper bound on this length generally leads to a better lower bound on the topological entropy. We address this and related problems in § 3 and establish some results (e.g. Lemma 3.8), which might also be of independent interest. The proof of Theorem 1.1 is then given in § 4. Finally, let us outline the argument for Theorem 1.3, and refer to § 5 for details. If $\mathrm {T}^{\infty }(\alpha )>0$ and $g$ denotes the element represented by $\Gamma$ in $\pi _1(M, \Gamma (0))$, then for each element $\mathfrak {a} \otimes \mathfrak {b}$ in $\mathrm {Comp}(\mu (g))$ there is an $\mathcal {F}$-transverse self-intersection point $y$ of $\Gamma$ for which the shortcut at $y$ represents either $\mathfrak {a}$ or $\mathfrak {b}$, and one can deduce the existence of a periodic point of a class that is induced by $\mathfrak {a}$ or $\mathfrak {b}$. Moreover, one can create several shortcuts to an iterate of $\Gamma$. We show that for (sufficiently large) $n>0$ the loops $\Gamma '= \gamma _{\rho _1}\gamma _{\rho _2} \cdots \gamma _{\rho _n}$ are associated to periodic points of period $nq$, where $\Gamma '$ is a concatenation of $n$ loops $\gamma _{\rho _i}$, each of which either is equal to $\Gamma$ or is obtained from $\Gamma$ by a shortcut at some of the above intersection points $y$ with one additional assumption: the turns at the chosen intersection points have to be either all to the left or all to the right. These observations allow us to conclude the inequality $\mathrm {H}^{\infty }(f)\geq ({1}/{q})\mathrm {T}^{\infty }(\alpha )$.

1.2 Hamiltonian diffeomorphisms and persistence of topological entropy

We now turn to applications of Theorem 1.1 in the context of Hamiltonian diffeomorphisms.

Let us consider a closed symplectic manifold $(M, \omega )$ and denote by $\mathrm {Ham}(M, \omega )$ the group of Hamiltonian diffeomorphisms on $M$, that is, the group of those diffeomorphisms that are the time-$1$ map of the (time-dependent) flow generated by a Hamiltonian vector field $X_{H_t}$ of some $H:S^1 \times M \to \mathbb {R}$ (cf. § 6.1). The group $\mathrm {Ham}(M, \omega )$ carries a distinctive bi-invariant metric $d_{\mathrm {Hofer}}$, the Hofer metric, which plays a central role in the study of rigidity phenomena of Hamiltonian diffeomorphisms. The distance $d_{\mathrm {Hofer}}(\varphi,\psi )$ for any $\varphi,\psi \in \mathrm {Ham}(M, \omega )$ may be defined to be $\|\varphi \circ \psi ^{-1}\|_{\mathrm {Hofer}}$, where

\[ \|\theta\|_{\mathrm{Hofer}} := \inf_{H} \int_{0}^{1}{\Big(\max_M H_t - \min_M H_t\Big)} \, dt, \]

with the infimum taken over all Hamiltonian functions $H : [0,1] \times M \to \mathbb {R}$ whose associated Hamiltonian flow has $\theta$ as time-$1$ map. It follows directly from the definition that small perturbations in the sense of Hofer's metric are not necessarily small in the $C^0$ metric.Footnote ³ The geometry of $(\mathrm {Ham}(M, \omega ), d_{\mathrm {Hofer}})$ and its interplay with dynamics has been thoroughly studied since its discovery by Hofer and his work in the early 1990s (see [Reference PolterovichPol01] for an extensive account with many results and references).

Hofer's metric plays an important role in the stability features of Floer homology. Let us briefly explain this in our context (more details can be found in § 6.1). Given that $(M, \omega )$ is symplectically aspherical and atoroidal, then for a free homotopy class $\alpha$ the action functional of a Hamiltonian $H$ on the smooth representatives $\mathcal {L}^{\alpha }(M)$ of $\alpha$ is given by $\mathcal {A}_H(x) = \int _0^1 H(t,x(t)) \,dt - \int _{\bar {x}} \omega$, where $\bar {x}:S^1 \times [0,1] \to M$ is a smooth map from an annulus to $M$, where one boundary component $S^1\times \{1\}$ maps to $x$ and the other $S^1 \times \{0\}$ to a fixed representative of $\alpha$. If $H$ is non-degenerate one defines the Floer homology as the homology of a chain complex generated by $1$-periodic orbits of $X_{H_t}$, and chain maps given by counting zero-dimensional moduli spaces of solutions $u:\mathbb {R} \times S^1 \to M$ of a certain perturbed nonlinear Cauchy–Riemann equation ${\bar {\partial }_{H,J}(u)=0}$ whose asymptotes $\lim _{s\to \pm \infty } u(s,t) = x_{\pm }(t)$ are $1$-periodic orbits $x_{\pm }(t)$ of $X_{H_t}$. Since action decreases along such solutions, the Floer homology can be filtered by action, taking into account only periodic orbits of action less than $a$, for varying $a \in \mathbb {R}$. The full Floer homology for a non-trivial free homotopy class $\alpha$ vanishes, but one can take the filtration into account in order to understand the structure and existence of orbits in $\alpha$. An elegant and fruitful way to keep track of the filtered Floer homology is to use the theory of persistent modules and barcodes [Reference Polterovich, Rosen, Samvelyan and ZhangPRSZ20]. With this terminology, any Hamiltonian gives rise to a barcode (i.e. a multiset of intervals in $\mathbb {R}$). The barcode, in this setting, only depends on the Hamiltonian diffeomorphism that is the time-$1$ map of $X_{H_t}$. Action estimates on continuation maps in Floer homology lead to stability properties of barcodes with respect to Hofer's metric, and hence in particular to the persistence of certain fixed points.

A special family $\mathcal {E}_g$ of Hamiltonian diffeomorphisms on surfaces $M=\Sigma _g$ of genus $g$, called eggbeaters, was introduced by Franjione and Ottino in [Reference Franjione and OttinoFO92], and used in a symplectic setting by Polterovich and Shelukhin in [Reference Polterovich and ShelukhinPS16]. The construction was carried out for surfaces of genus $g\geq 4$ and later extended by the first author in [Reference ChorCho22] to surfaces of genus $g\geq 2$. Computations of certain barcode invariants were carried out and it was proved that $\mathcal {E}_g$ provide a class of Hamiltonian diffeomorphisms whose Hofer distance to the space of autonomous Hamiltonian diffeomorphisms can be arbitrarily large. The same results pass to some products $\Sigma _g\times N$ [Reference Polterovich and ShelukhinPS16, Reference ZhangZha19], and the constructions allow the free group of two generators to be embedded into the asymptotic cone of the group of Hamiltonian diffeomorphisms equipped with Hofer's metric [Reference Alvarez-Gavela, Kaminker, Kislev, Kliakhandler, Pavlichenko, Rigolli, Rosen, Shabtai, Stevenson and ZhangAGK⁺19, Reference ChorCho22].

One may formulate the above results as (a consequence of) persistence of certain dynamical properties of eggbeaters. Eggbeaters are prototypical for a chaotic dynamical system. While this chaotic behavior, as opposed to integrable behavior, served as motivation to investigate these maps in [Reference Polterovich and ShelukhinPS16], the question of the persistence of chaos, that is, the persistence of entropy, exponential orbit complexity etc., has not yet been addressed. The following result gives some answers in the setting of surfaces of genus $g\geq 2$.

Theorem 1.4 Let $(\Sigma _g,\sigma _g)$ be a surface of genus $g\geq 2$ with an area form $\sigma _g$. There is a sequence $\phi _l \in \mathcal {E}_g$ of eggbeaters on $\Sigma _g$ with $M_l := \|\phi _l\|_{\mathrm {Hofer}} \to \infty$ and constants $\delta, C>0$, such that for all $l \in \mathbb {N}$ and all $\psi \in \mathrm {Ham}(M,\omega )$ with $d_{\mathrm {Hofer}}(\psi, \phi _l) < \delta M_l$,

\[ \mathrm{H}^{\infty}(\psi) \geq \log(C M_l^2) \]

and, in particular,

(1)\begin{equation} h_{\mathrm{top}}(\psi) \geq \log(C M_l^2). \end{equation}

In particular, using the terminology at the beginning of this introduction, for any constant $E$ there is an unbounded family in $(\mathrm {Ham}(\Sigma _g,\sigma _g), d_{\mathrm {Hofer}})$ on which (‘$h_{\mathrm {top}}\geq E$’) $\delta$-persists.

The sequence $\phi _l$ satisfies $h_{\mathrm {top}}(\phi _l) \leq \log (C'M_l^2)$ for some constant $C' > C$ (see Remark 7.4), and hence the lower bounds in (1) are ‘optimal’ from an asymptotic viewpoint, or, in other words, eggbeaters are ‘almost’ minimal points for the topological entropy on $\mathrm {Ham}(\Sigma _g,\sigma _g)$. More precisely, we shall prove the following corollary in § 7.

Corollary 1.5 Let $(\Sigma _g,\sigma _g)$ be as above. Then there are constants $K>0$ and $\delta >0$, and a sequence of $\phi _l \in \mathcal {E}_g$ with $M_l := \|\phi _l\|_{\mathrm {Hofer}} \to \infty$ and $h_{\mathrm {top}}(\phi _l) \to \infty$, such that all $\psi \in \mathrm {Ham}(\Sigma _g, \sigma _g)$ with $d(\psi, \phi _l) < \delta M_l$ satisfy

\[ h_{\mathrm{top}}(\psi) \geq h_{\mathrm{top}}(\phi_l) - K. \]

The sequence $\phi _l$ defines an element in the asymptotic cone of $(\mathrm {Ham}(\Sigma _g, \sigma _g), d_{\mathrm {Hofer}})$. An interesting question is whether similar minimality properties hold also for defining sequences of most elements in the image of the embedding of $F_2$ into the asymptotic cone of $(\mathrm {Ham}(\Sigma _g, \sigma _g), d_{\mathrm {Hofer}})$ obtained in [Reference Alvarez-Gavela, Kaminker, Kislev, Kliakhandler, Pavlichenko, Rigolli, Rosen, Shabtai, Stevenson and ZhangAGK⁺19], and this question motivates efforts to further strengthen the bounds obtained in Theorem 1.3.

Finally, we will also observe that the lower bounds obtained in Theorem 1.4 are Hofer generic and we show the following theorem.

The proofs of Theorems 1.4 and 1.6 and Corollary 1.5 are given in § 7.

A recent result in [Reference KhanevskyKha21], building on different methods (cf. [Reference Brandenbursky and MarcinkowskiBM19]), asserts that for surfaces $\Sigma _g$ of genus $g\geq 1$ there is for any $C\geq 0$ a class of Lagrangian pairs $(L,L')$, $L,L'\subset \Sigma _g$, such that $h_{\mathrm {top}}(\varphi ) > C$ for all $\varphi$ with $\varphi (L) = L'$. It would be interesting to understand whether these sets of pairs may satisfy some stability properties similar to those of Theorems 1.4 and 1.6.

Very recently, the authors of [Reference Çineli, Ginzburg and GürelÇGG21] show that the topological entropy of a Hamiltonian diffeomorphism $\varphi$ on a closed surface coincides with its barcode entropy $\hbar (\varphi )$ which they introduce, and which measures the growth of the number of certain bars in the barcode of the iterates of $\varphi$. Hence, together with the results in this paper, this shows that the results obtained in Theorems 1.4, 1.6 and Corollary 1.5 hold additionally for $\hbar$. This is noteworthy, since it is a priori not clear which stability properties hold for $\hbar$ with respect to Hofer's metric.

Also recently, motivated by the present work, stability properties with respect to $d_{\mathrm {Hofer}}$ on the braid types of periodic orbits of Hamiltonian surface diffeomorphisms have been studied by Alves and the second author (see [Reference Alves and MeiwesAM21]). One dynamical consequence is that $h_{\mathrm {top}}$ is lower semi-continuous on $(\mathrm {Ham}(\Sigma, \omega ), d_{\mathrm {Hofer}})$ for closed surfaces $\Sigma$.

Related questions of global robustness of positive entropy for families of contactomorphisms on contact manifolds were studied extensively and fruitfully in recent years by various methods. A large class of contactomorphisms are those that arise via Reeb flows, and there is an abundance of contact manifolds for which the topological entropy or the exponential orbit growth rate is positive for all Reeb flows. Examples and dynamical properties of those manifolds are investigated in [Reference Abbondandolo, Alves, Saglam and SchlenkAASS21, Reference AlvesAlv16a, Reference AlvesAlv16b, Reference AlvesAlv19, Reference Alves, Colin and HondaACH19, Reference Alves and MeiwesAM19, Reference CôtéCôt21, Reference Frauenfelder and SchlenkFS06, Reference Macarini and SchlenkMS11, Reference MeiwesMei18]. Some of these results generalize to positive contactomorphisms [Reference DahindenDah18], and results on the dependence of some lower bounds on the topological entropy with respect to their positive contact Hamiltonians have been obtained in [Reference DahindenDah21]. Forcing results for Reeb flows are obtained in [Reference Alves and PirnapasovAP22]. A related discussion and results on questions of $C^0$-stability of the topological entropy of geodesic flows can be found in [Reference Alves, Dahinden, Meiwes and MerlinADMM22]. While the approach in our paper is suited to dimension $2$, different methods yield higher-dimensional symplectic manifolds for which conclusions similar to that of Theorems 1.4 and 1.6 hold. This will be discussed elsewhere.

1.3 Structure of the paper

In § 2 we recall the theory of transverse foliations on surfaces, which is the setting in which Theorems 1.1–1.3 are proved. In § 3 we restrict our attention to loops with so-called $\mathcal {F}$-transverse self-intersections, and prove a few key claims.

In § 4 we prove Theorems 1.1 and 1.2, using the tools developed in the previous sections. In § 5 we define the growth rate $\mathrm {T}^{\infty }$ of a free homotopy class and prove Theorem 1.3.

Section 6 gives a short background on Floer theory and persistence modules, which is then used in § 7 to derive bounds on entropy and periodic orbit growth of large perturbations with respect to Hofer's metric of eggbeater maps. This will, in particular, prove Theorems 1.4 and 1.6.

2.1 Surface foliations and transverse paths

In the following let $M$ be an oriented surface. The plane $\mathbb {R}^2$ will be endowed with the usual orientation. A path on $M$ is a continuous map $\gamma :J \to M$, defined on an interval $J\subset \mathbb {R}$. A path $\gamma$ is proper if $J$ is open and the preimage of every compact subset of $M$ is compact. A line is an injective and proper path $\lambda : J \to M$; it inherits a natural orientation induced by the usual orientation of $\mathbb {R}$. If $M=\mathbb {R}^2$, the complement of $\lambda$ has two connected components, one on the right, $R(\lambda )$, and one on the left, $L(\lambda )$. A loop is a continuous map $\Gamma :S^1 = \mathbb {R}/ \mathbb {Z} \to M$. It lifts to a path $\gamma : \mathbb {R} \to M$ with $\gamma (t+1) = \gamma (t)$ for all $t\in \mathbb {R}$, the natural lift of $\Gamma$. For two closed finite intervals $J =[a,b],J'=[a',b']$ and paths $\gamma :J \to M$, $\gamma ':J' \to M$ with $\gamma (b) = \gamma '(a')$, we denote by $\gamma \gamma '$ the usual concatenation of paths. In particular, $\Gamma ^m$, for $m\in \mathbb {N}$, is the $m$-fold iteration of a loop $\Gamma$ (i.e. $\Gamma ^m(t) := \gamma (mt)$). The path obtained by reverse parametrization of $\gamma$ is denoted by $\overline {\gamma }$.

A singular oriented foliation on $M$ is an oriented topological foliation $\mathcal {F}$ defined on an open set of $M$. This set is called the domain of $\mathcal {F}$ and is denoted by $\mathrm {dom} (\mathcal {F})$. The complement $M \setminus \mathrm {dom} (\mathcal {F})$ is the singular set, denoted by $\mathrm {sing} (\mathcal {F})$. We denote by $\phi _z$ the leaf passing through $z\in \mathrm {dom} (\mathcal {F})$, and by $\phi _z^+$ the positive and by $\phi _z^-$ the negative half-leaf. A path $\gamma : J \to M$ is (positively) transverse to $\mathcal {F}$ or $\mathcal {F}$-transverse if its image does not meet the singular set and if, for every $t_0 \in J$, there is a continuous chart $h: W \to (0,1)^2$ at $\gamma (t_0)$ compatible with the orientation and sending the foliation $\mathcal {F}|_{W}$ onto the vertical foliation oriented downwards such that the map $\pi _1 \circ h \circ \gamma$ is increasing in a neighborhood of $t_0$, where $\pi _1$ is the vertical projection. If $\mathrm {dom}(\mathcal {F})$ is connected we denote by $\widetilde {\mathrm {dom}(\mathcal {F})}$ the universal covering space of the surface $\mathrm {dom}(\mathcal {F})$, otherwise we denote by $\widetilde {\mathrm {dom}(\mathcal {F})}$ the disjoint union of the universal coverings of its connected components. Then $\mathcal {F}|_{\mathrm {dom} (\mathcal {F})}$ lifts to a (non-singular) foliation $\widetilde {\mathcal {F}}$ on $\widetilde {\mathrm {dom}(\mathcal {F})}$. Note that since there is no non-singular foliation on $S^2$, $\widetilde {\mathrm {dom}(\mathcal {F})}$ is always homeomorphic to $\mathbb {R}^2$, or to several copies of $\mathbb {R}^2$ if $\mathrm {dom}(\mathcal {F})$ is disconnected. Every lift of an $\mathcal {F}$-transverse path $\gamma$ is an $\widetilde {\mathcal {F}}$-transverse path on $\widetilde {\mathrm {dom}(\mathcal {F})}$. We say that $\widetilde {\gamma }: \mathbb {R} \to \widetilde {\mathrm {dom}(\mathcal {F})}$ is a lift of the loop $\Gamma$ (to $\widetilde {\mathrm {dom}(\mathcal {F})}$) if it is a lift of its natural lift $\gamma : \mathbb {R} \to \mathrm {dom}(\mathcal {F})$.

If $\mathcal {F}$ is a non-singular foliation on $\mathbb {R}^2$, then two $\mathcal {F}$-transverse paths $\gamma : J \to \mathbb {R}^2$ and $\gamma ': J'\to \mathbb {R}^2$ are equivalent for $\mathcal {F}$ if there exists an increasing homeomorphism $h:J \to J'$ such that $\phi _{\gamma '(h(t))} = \phi _{\gamma (t)}$ for every $t\in \mathbb {R}$. In general, if $\mathcal {F}$ is a (possibly singular) foliation on an oriented surface $M$, two transverse paths $\gamma : J \to M$ and $\gamma ': J'\to M$ are called equivalent for $\mathcal {F}$ if they can be lifted to the universal covering space $\widetilde {\mathrm {dom}{(\mathcal {F})}}$ of $\mathrm {dom}(\mathcal {F})$ as paths that are equivalent for the lifted foliation $\widetilde {\mathcal {F}}$. A loop $\Gamma : S^1 \to \mathrm {dom}(\mathcal {F})$ is called positively transverse to $\mathcal {F}$ if this holds for the natural lift $\gamma : \mathbb {R} \to \mathrm {dom}(\mathcal {F})$. Two $\mathcal {F}$-transverse loops $\Gamma$ and $\Gamma '$ are equivalent if there exist two lifts $\widetilde {\gamma }: \mathbb {R} \to \widetilde {\mathrm {dom}(\mathcal {F})}$ and $\widetilde {\gamma }': \mathbb {R} \to \widetilde {\mathrm {dom}(\mathcal {F})}$ of $\Gamma$ and $\Gamma '$ respectively, a deck transformation $T$ and an orientation-preserving homeomorphism $h:\mathbb {R} \to \mathbb {R}$ invariant by $t\mapsto t+1$ and such that for all $t\in \mathbb {R}$,

\begin{align*} \widetilde{\gamma}(t+1) = T(\widetilde{\gamma}(t)), \quad \widetilde{\gamma}'(t+1) = T(\widetilde{\gamma}'(t)), \quad \phi_{\widetilde{\gamma}'(h(t))} = \phi_{\widetilde{\gamma}(t)}. \end{align*}

2.2 $\mathcal {F}$-transverse intersection

We now recall the definition of $\mathcal {F}$-transverse intersection, which is a central notion in [Reference Le Calvez and TalLeCT18]. Many details and illustrating figures can also be found in [Reference Le Calvez and TalLeCT18, §§ 2 and 3].

Let $\lambda _0, \lambda _1$, and $\lambda _2$ be three lines in $\mathbb {R}^2$. The line $\lambda _2$ is above $\lambda _1$ relative to $\lambda _0$ (or $\lambda _1$ is below $\lambda _2$ relative to $\lambda _0$) if:

• • the three lines are pairwise disjoint;

• • none of the lines separates the two others;

• • if $\gamma _1, \gamma _2$ are two disjoint paths that join $z_1=\lambda _0(t_1)$ (respectively, $z_2 = \lambda _0(t_2)$) to $z'_1\in \lambda _1$ (respectively, $z'_2 \in \lambda _2$) and do not meet the three lines except at the ends, then $t_2 >t_1$.

Figure 1.

$\gamma _1$ and $\gamma _2$ intersect $\mathcal {F}$-transversally and positively at $\phi$.

Assume $\mathcal {F}$ is a non-singular foliation on $\mathbb {R}^2$. Let $\gamma _1: J_1 \to \mathbb {R}^2$ and $\gamma _2: J_2 \to \mathbb {R}^2$ be two transverse paths such that $\phi _{\gamma _1(t_1)} = \phi _{\gamma _2(t_2)} = \phi$. The paths $\gamma _1$ and $\gamma _2$ intersect $\mathcal {F}$-transversally and positively at $\phi$ (see Figure 1) if there exist $a_1, b_1$ in $J_1$ and $a_2, b_2 \in J_2$ satisfying $a_1 < t_1 < b_1$, $a_2 < t_2 < b_2$, such that

• • $\phi _{\gamma _2(a_2)}$ is below $\phi _{\gamma _1(a_1)}$ relative to $\phi$, and

• • $\phi _{\gamma _2(b_2)}$ is above $\phi _{\gamma _1(b_1)}$ relative to $\phi$.

In this situation, we also say that $\gamma _1$ intersects $\gamma _2$ $\mathcal {F}$-transversally and positively, $\gamma _2$ and $\gamma _1$ intersect $\mathcal {F}$-transversally and negatively, or $\gamma _2$ intersects $\gamma _1$ $\mathcal {F}$-transversally and negatively at $\phi$.

Now let $\mathcal {F}$ be a (possibly singular) foliation on an oriented surface $M$. Let $\gamma _1: J_1 \to M$ and $\gamma _2: J_2 \to M$ be two transverse paths such that $\phi _{\gamma _1(t_1)} = \phi _{\gamma _2(t_2)} = \phi$. We say that $\gamma _1$ and $\gamma _2$ intersect $\mathcal {F}$-transversally and positively at $\phi$ (respectively, negatively at $\phi$) if there exist paths $\widetilde {\gamma }_1: J_1 \to \widetilde {\mathrm {dom}(\mathcal {F})}$ and $\widetilde {\gamma }_2: J_2 \to \widetilde {\mathrm {dom}(\mathcal {F})}$, lifting $\gamma _1$ and $\gamma _2$, with a common leaf $\widetilde {\phi } = \phi _{\widetilde {\gamma }_1(t_1)} = \phi _{\widetilde {\gamma }_2(t_2)}$ that lifts $\phi$ such that $\widetilde {\gamma }_1$ and $\widetilde {\gamma }_2$ intersect $\widetilde {\mathcal {F}}$-transversally and positively at $\widetilde {\phi }$ (respectively, negatively at $\widetilde {\phi }$). If two paths $\gamma _1$ and $\gamma _2$ intersect $\mathcal {F}$-transversally, there exist $t_1'$ and $t_2'$ such that $\gamma _1(t_1') = \gamma _2(t_2')$ and such that $\gamma _1$ and $\gamma _2$ intersect $\mathcal {F}$-transversally at $\phi _{\gamma _1(t_1')} = \phi _{\gamma _2(t_2')}$. We say that $\gamma _1$ and $\gamma _2$ intersect $\mathcal {F}$-transversally at $\gamma _1(t_1') = \gamma _2(t_2')$. A transverse path $\gamma$ has a (positive) $\mathcal {F}$-transverse self-intersection at $\gamma (t_1) = \gamma (t_2)$, $t_1 < t_2$, if for every lift $\widetilde {\gamma }$ there is a deck transformation $U$ such that $\widetilde {\gamma }$ and $U\widetilde {\gamma }$ have a (positive) $\widetilde {\mathcal {F}}$-transverse intersection at $\widetilde {\gamma }(t_1) = U\widetilde {\gamma }(t_2)$. A transverse loop $\Gamma$ has an $\mathcal {F}$-transverse self-intersection at $\Gamma (t_1) = \Gamma (t_2)$ if its natural lift $\gamma$ has an $\mathcal {F}$-transverse self-intersection at $\gamma (t_1) = \gamma (t_2)$.

Finally, we say that a transverse path $\gamma :[a,b] \to \mathbb {R}^2$ (for a regular foliation $\mathcal {F}$ on $\mathbb {R}^2$) has a leaf on its right (respectively, a leaf on its left), if there is a leaf $\phi$ such that $\phi$ is above (respectively, below) $\phi _{\gamma (a)}$ relative to $\phi _{\gamma (b)}$. We say that $\gamma :[a,b] \to M$ has a leaf on its right (respectively, a leaf on its left), if a lift of $\gamma$ to $\widetilde {\mathcal {F}}$ has a leaf on its right (respectively, a leaf on its left).

2.3 Identity isotopies

In the following, let $f$ be a homeomorphism on $M$ that is isotopic to the identity. Let $\mathcal {I}$ be the set of isotopies $I = (f_t)_{t\in [0,1]}$ between the identity and $f$. Here, isotopy means a continuous path of homeomorphisms with respect to the topology defined by the uniform convergence of maps and their inverses on compact sets. For $I \in \mathcal {I}$, the trajectory $I(z)$ of a point $z\in M$ is defined to be the path $t \mapsto f_t(z)$.

Let $\mathrm {Fix}(I) := \bigcap _{t\in [0,1]} \mathrm {Fix}(f_t)$ and $\mathrm {dom} (I) = M \setminus \mathrm {Fix}(I)$. There is the following preorder on $\mathcal {I}$. We define $I < I'$ if

• • $\mathrm {Fix}(I) \subset \mathrm {Fix}(I')$, and

• • $I'$ is homotopic to $I$ relative to $\mathrm {Fix}(I)$.

For each $I \in \mathcal {I}$, there is $I'\in \mathcal {I}$ with $I< I'$ and such that $I'$ is maximal with respect to <. This was proved in [Reference JaulentJau14] with certain restrictions and in [Reference Béguin, Crovisier and Le RouxBCLR20] in full generality. See also [Reference Humilière, Le Roux and SeyfaddiniHLRS16] for the case of diffeomorphisms. Maximal elements are exactly those $I \in \mathcal {I}$ such that for every $z\in \mathrm {Fix}(f) \setminus \mathrm {Fix}(I)$ the loop $I(z)$ is not contractible in $\mathrm {dom}(I)$ (see [Reference JaulentJau14]). A foliation $\mathcal {F}$ on $M$ is called transverse to I if

• • the singular set $\mathrm {sing}(\mathcal {F})$ coincides with $\mathrm {Fix}(I)$, and

• • for every $z\in \mathrm {dom}(I)$, the trajectory $I(z)$ is homotopic in $\mathrm {dom}(I)$ relative to the endpoints to a path $\gamma$ positively transverse to $\mathcal {F}$.

One has the following fundamental result.

2.4 Admissible paths

Let $I$ be a maximal isotopy to $f$ and $\mathcal {F}$ be transverse to $I$. Let $I_{\mathcal {F}}(z)$ denote the class of paths that are positively transverse to $\mathcal {F}$, that join $z$ and $f(z)$, and that are homotopic in $\mathrm {dom}(I)$ to $I(z)$ relative to the endpoints. Every path in this class is called a transverse trajectory to $z$. One defines for every $n$, $I_{\mathcal {F}}^n(z) := \Pi _{0\leq k < n} I_{\mathcal {F}}(f^k(z))$, which means the class of paths that can be written as concatenation of paths in $I_{\mathcal {F}}(f^k(z))$, $0\leq k < n$.

A path $\gamma :[a,b] \to \mathrm {dom}(I)$ positively transverse to $\mathcal {F}$ is called admissible of order $n$ if it is equivalent to a path in $I^n_{\mathcal {F}}(z)$ for some $z\in \mathrm {dom}(I)$. A path $\gamma$ is called admissible of order at most $n$ if it is a subpath of a path that is admissible of order $n$. A transverse path that has a leaf on its right or a leaf on its left and that is admissible of order at most $n$, is also admissible of order $n$ [Reference Le Calvez and TalLeCT18, Proposition 19].

The concept of $\mathcal {F}$-transverse intersections allows us to show admissibility for various transverse paths. Let us state the ‘fundamental proposition’ in [Reference Le Calvez and TalLeCT18].

This proposition is essential for the following result.

The second statement of Proposition 2.4 is a direct consequence of Proposition 2.3, whereas the first statement follows from Proposition 2.3 and from the fact that there is $q$ such that $\gamma |_{[a,s]}(\gamma |_{[s,t]})^q \gamma |_{[t,b]}$ is not admissible of order $n$ (see [Reference Le Calvez and TalLeCT18, Proof of Proposition 23]). Besides Proposition 2.4 we will use the following, slightly more general, statement.

Certain assumptions on a transverse loop $\Gamma$ guarantee the existence of periodic points of $f$. An $\mathcal {F}$-transverse loop $\Gamma$ with natural lift $\gamma$ is linearly admissible of order $q$ if there exist two sequences $(r_k)_{k\geq 0}$ and $(s_k)_{k\geq 0}$ of natural numbers with

\begin{align*} &\lim_{k\to \infty} r_k = \lim_{k\to \infty} s_k = +\infty, \quad \limsup_{k\to \infty} r_k/s_k \geq 1/q,\\ &\text{and such that }\gamma|_{[0,r_k]} \text{ is admissible of order at most } s_k. \end{align*}

If $z$ is a periodic point of period $q$, then there exists a transverse loop $\Gamma '$ whose natural lift satisfies $\gamma '|_{[0,1]} = I_{\mathcal {F}}^q(z)$. A transverse loop $\Gamma$ is said to be associated to $z$ if it is $\mathcal {F}$-equivalent to $\Gamma '$. Note that $\Gamma$ is then linearly admissible of order $q$. The following important realization result asserts a partial converse.