Entropy and periodic points in pseudo-Anosov isotopy classes

Introduction

Let $M$ be a closed connected orientable surface of genus at least two. The Nielsen–Thurston classification [29] describes its mapping classes in terms of periodic, reducible, and pseudo-Anosov representatives. Periodic representatives have finite order. Reducible representatives preserve a nonempty finite collection of disjoint essential simple closed curves; after cutting along a suitable such collection and passing to an iterate, the induced mapping classes on the complementary components are periodic or pseudo-Anosov. The pseudo-Anosov pieces are therefore the source of positive entropy in the canonical decomposition. Their importance for dynamics lies in the complexity they impose on every representative of the isotopy class.

We fix a pseudo-Anosov homeomorphism $f:M\to M$. It preserves two transverse singular measured foliations, multiplying their transverse measures by $\lambda$ and $\lambda^{-1}$, where $\lambda>1$ is its stretch factor. This expansion and contraction determine its entropy, which is minimal in its isotopy class:

Thus the isotopy class already requires positive entropy, while its representatives may have additional dynamical complexity.

The complexity of map is typically related to the presence of periodic points. In the case of surfaces, the powerful Nielsen’s fixed point theory [21] provides a way to group periodic points according to some topological criterion. Let $g\simeq f$ and $n\geq1$ where, from now on, $g\simeq f$ means that $g$ is isotopic to $f$.

The resulting equivalence classes are the Nielsen classes of $g^n$. We denote

Key words and phrases. Pseudo-Anosov homeomorphism, Nielsen classes, entropy of maximal entropy.

P.D. Carrasco was partially supported by CNPq (Produtividade em Pesquisa) 305197/2025-8, Projeto Universal CNPq 401737/2025-0, FAPEMIG APQ-02338-24, and CAPES through the MATH-AMSUD program.

F. Rodriguez-Hertz was partially supported by NSF grant DMS-2453688. and define

We count every nonempty class. Our purpose is to determine how much of the dynamical complexity of $g$ is detected by the growth of these classes, and when this growth captures its full topological entropy.

The basic bounds are

Handel’s results [13 Corollary 3.6] characterize the minimum: $h_N(g)=h_{\operatorname{top}}(f)$ precisely when $g$ admits a semiconjugacy onto $f$ homotopic to the identity. To compare Nielsen growth with the entropy of other representatives, we use Fathi’s hyperbolic extension [10] and the global orbit quotients constructed in [23]. These quotients place the dynamics of every representative inside a common ambient system.

Let $\widetilde{M}$ be the universal cover of $M$, with deck group $\Gamma$, and fix a lift $F$ of $f$. Collapsing each leaf of the lifted stable or unstable foliation gives a real tree $T^s$ or $T^u$, respectively, whose metric is induced by the transverse measure. Write $\widehat{T}^s$ and $\widehat{T}^u$ for their metric completions, and let

where $\Gamma$ acts diagonally. The lift $F$ induces a product homeomorphism $\widetilde{\Phi}$ of $T$, expanding distances in $\widehat{T}^s$ by $\lambda$ and contracting distances in $\widehat{T}^u$ by $\lambda^{-1}$. The superscript records the foliation being collapsed: the stable-leaf tree is the expanding coordinate. The product map descends to a homeomorphism $\Phi:Z\to Z$. The pairs of intersecting stable and unstable leaves determine an embedded copy $L(f)$ of $M$ in $Z$, and $\left.\Phi\right|_{L(f)}$ is conjugate to $f$.

For $g\simeq f$, choose the lift $G$ obtained by lifting an isotopy from $f$ to $g$, starting at $F$.

Thus this relation identifies points whose lifted orbits remain at bounded distance in both time directions. Fathi’s construction gives a continuous surjection $H_g:M\to L(g)$ onto a compact $\Phi$-invariant subset of $Z$ such that

The last three authors of this article identified the fibers of $H_g$ precisely [23]:

It follows that $L(g)$ realizes the quotient $M/\sim_g$, and $\Phi_g$ realizes its induced dynamics. The ambient map $\Phi$ is fixed throughout the isotopy class; the representative $g$ determines the invariant set $L(g)$ on which it is restricted.

The inclusion $L(f)\subset L(g)$ expresses the persistence of the pseudo-Anosov dynamics under global shadowing. Handel [12] proved that every $g\simeq f$ has a compact invariant subset whose dynamics factors onto $f$; in this construction, that subset is

The complement $L(g)\setminus L(f)$ records the global orbit classes that do not correspond to orbits of $f$.

For periodic points, the relation with Nielsen classes is particularly direct. If $p,q\in\operatorname{Fix}(g^n)$, then $H_g(p)=H_g(q)$ if and only if $p$ and $q$ are Nielsen equivalent [19 Lemma 2.8]. Thus each nonempty Nielsen class of $g^n$ determines a point of $\operatorname{Fix}(\Phi_g^n)$, and distinct classes determine distinct points.

Our first theorem identifies the exponential growth rates of these two counts with the entropy of the quotient.

This answers [23 Question 4.2] under the stated regularity hypotheses. In particular, the quotient satisfies Bowen’s periodic-point entropy formula. Although $\Phi_g$ is expansive, this formula requires an argument beyond the hyperbolicity of the ambient extension: the sets $L(g)$ are not locally maximal, and the ambient shadowing property does not pass to their restrictions.

Our second theorem compares the entropy of the quotient with that of the surface dynamics. The geometry of $L(g)$ is essential here. These spaces need not be surfaces: their local stable and unstable components are dendrites, that is, locally connected continua containing no simple closed curve. Every periodic point outside $L(f)$ is an endpoint of both components [23]. We use this endpoint geometry together with the recent method developed by the authors [9], that relates entropy and semiconjugacy on surfaces. We emphasize that several additional difficulties appear while dealing with global orbit quotients, due to lack of any differentiable structure.

Thus, whenever the quotient entropy exceeds the pseudo-Anosov minimum, it is strictly smaller than the entropy of the representative. Together with Theorem 1.3 and the basic entropy bounds, this shows that Nielsen growth captures the full entropy of a smooth representative precisely when that representative minimizes entropy in its isotopy class. We record this consequence as Theorem 4.2 in § 4.

Our third theorem concerns the dependence of quotient entropy on the representative.

No differentiability is required of the approximating homeomorphisms $g_j$.

Quotient entropy therefore refines the pseudo-Anosov entropy bound while remaining stable, at these representatives, under perturbations that impose no control on derivatives. This answers the continuity part of [23 Question 4.1] at $C^{1+\alpha}$ diffeomorphisms. Together with Theorem 1.3, this implies that if $g_j$ and $g$ are $C^{1+\alpha}$ diffeomorphisms isotopic to $f$ and $g_j \to g$ uniformly, then $h_N(g_j) \to h_N(g)$.

The invariant foliations of $f$ also connect these entropy questions with Teichmüller theory. They are the horizontal and vertical foliations of a quadratic differential and determine a geodesic in Teichmüller space, the space of marked conformal structures on $M$. The mapping class of $f$ translates along this geodesic by $\log\lambda$ [4]. The minimum entropy in the isotopy class thus has a geometric interpretation: it is the translation length of the class on Teichmüller space. The projective classes of the two foliations are its two fixed points on Thurston’s boundary.

These same foliations give the dual trees used in this paper. The translation lengths of the deck transformations on each tree record intersection numbers of closed curves with the corresponding measured foliation. Such tree actions belong to the geometric picture developed by Morgan and Shalen [20], in which real trees describe degenerations of hyperbolic structures. Fathi’s construction [10] brings the two trees together in the hyperbolic system $(Z,\Phi)$, and the sets $L(g)$ of [23] realize the global orbit quotients of representatives throughout the isotopy class inside this common space. The geometry supplied by $f$ therefore also allows us to study the additional complexity of $g$. Our results compare the entropy retained by this quotient with both the geometric minimum $\log\lambda$ and the full entropy of the representative.

We note that in a recent preprint of García-Sassi and Tal [11], the authors consider a related quotient construction for maps in the isotopy class of a pseudo-Anosov, by collapsing maximal continua with bounded infinite orbit in the universal cover. The resulting quotient map sits above Fathi’s construction, and is CW-expansive. It seems very likely that some methods in this paper can be applied in that situation; investigating such extent remains as an interesting open problem.

The proofs proceed as follows. § 2 recalls the dual trees, Fathi’s extension, and the realization of the global orbit quotient. § 3 collects the preliminaries on hyperbolic measures, homoclinic classes, conditional measures, and horseshoes.

§ 4 proves Theorems 1.3 and 1.4 and derives Corollary 4.2. For the entropy formula, the correspondence between Nielsen classes and images of periodic points, together with expansivity, gives one inequality. For the reverse inequality, we lift positive-entropy quotient measures to hyperbolic measures on the surface. The Brin–Katok formula [6] supplies many return segments whose images in the quotient are separated; Katok’s closing lemma [15] turns these segments into periodic points belonging to distinct Nielsen classes.

The proof of the strict gap develops the geometric argument of [9]. Under entropy equality above $h_{\operatorname{top}}(f)$, a maximal-entropy measure of the quotient lifts to a hyperbolic maximal-entropy measure with zero relative entropy over $H_g$. The conditional-measure argument recalled in § 3.3 then gives injectivity on full conditional-measure subsets of stable and unstable manifolds. Homoclinic periodic approximation provides a saddle projecting outside $L(f)$, where the endpoint geometry forces a fold in the tree image of its unstable manifold. The HHTU Sard argument [25], in its adaptation to maximal-entropy measures [8], supplies transverse stable crossings of the folded arc. Ben Ovadia’s absolute continuity result [3] allows these crossings to be chosen in the set of injectivity, giving a contradiction. The $C^\infty$ hypothesis enters through the Sard argument.

§ 5 proves Theorem 1.5. Upper semicontinuity follows from the Hausdorff continuity of $L(g)$ inside the common expansive ambient system. For lower semicontinuity, we incorporate returns separated in the quotient into the Katok–Mendoza construction, obtaining locally maximal horseshoes on which $H_g$ is injective. Nitecki’s $C^0$ semistability theorem [22], together with expansivity, ensures that their images in the quotient persist under nearby homeomorphisms.

§ 6 presents the remaining questions.

For convenience of the reader we reproduce the unpublished article [23] in the Appendix.

Dual trees and the universal hyperbolic extension

Let $M$ be a closed connected orientable surface of genus at least two, and let $f \colon M \to M$ be a pseudo-Anosov homeomorphism with dilatation $\lambda> 1$. We denote by $\mathcal{F}^{s},\mathcal{F}^{u}$ its stable and unstable measured foliations, respectively. Fix a universal covering $p \colon\widetilde{M} \to M$, with deck group $\Gamma$, and a lift $F \colon\widetilde{M} \to\widetilde{M}$ of $f$.

We recall Fathi’s hyperbolic extension [10] and the realization of the orbit quotients in [23].

The dual trees

For $\sigma\in\{s,u\}$, let