Surface homeomorphisms with big rotation set

TL;DR

This paper develops a structural theory for surface homeomorphisms of genus $g\ge 2$ with big homological rotation sets, proving that the rotation set $\operatorname{rot}(f)$ is a finite union of convex pieces containing $0$, with the number of pieces bounded by $\max(g,\lfloor g^2/4\rfloor)$. It introduces a graph- and Markovian-intersection framework (graphs $G$ and $\mathcal{T}$) that encodes rotational interactions, enabling invariant open decompositions of the surface where each part realizes a distinct rotation sub-collection, while the remaining region carries zero rotation. A two-pronged bounded-deviation analysis shows that homological deviations are uniformly bounded with respect to $\operatorname{rot}(f)$ and with respect to the convex hull $\operatorname{conv}(\operatorname{rot}(f))$, yielding realization results for rotation vectors and Boyland-type consequences in this higher-genus setting. The paper also clarifies continuity properties of rotation-set maps, proving openness of the big-rotation-set class and providing conditions under which ergodic and ordinary rotation-sets vary continuously, alongside explicit counterexamples illustrating the limits of semicontinuity.

Abstract

This article consists in applications of [arXiv:2511.14232] in the case of homemomorphisms of higher genus surfaces whose homological rotation set is big enough -- a class of dynamics that is open. We first prove a structure theorem for the rotation set of such homeomorphisms: it is a finite union of convex sets, we get an optimal bound for the number of such pieces. This bound can be improved in the case of transitive (in this case the rotation set is convex) and non-wandering dynamics (and for such homeomorphisms we get the existence of a family of invariant essential open sets). We also get boundedness of deviations for homeomorphisms with big rotation set and some consequences of it, including a answer to Boyland's conjecture in our framework.

Figures (5)

• Figure 1: The space $\operatorname{Homeo}_0(S)$ from a rotational viewpoint. Here $\lambda$ is Lebesgue measure. We have the inclusions $\{h_{top} = 0\}\subset \{\mathrm{dim}( \mathrm{span}(\operatorname{rot})) \leq g\}$ and $\{\mathrm{int}(\operatorname{rot}_{\mathrm{erg}}) \neq\emptyset\}\subset \{\mathrm{int}(\operatorname{conv}(\operatorname{rot}_{\mathrm{erg}})) \neq\emptyset\}\subset \{\mathrm{dim}( \mathrm{span}(\operatorname{rot})) > g\}$. The dots represent the set $\{h_{top} = 0\}$. The present work is focused on the underlined case $\mathrm{int}(\operatorname{conv}(\operatorname{rot}_{\mathrm{erg}})) \neq\emptyset$. Note that an open and dense subset of $\operatorname{Homeo}_0(S)$ was studied in depth from a rotational viewpoint in MR4578317.
• Figure 2: An example of homeomorphism of $\operatorname{Homeo}_0(S)$ with big rotation set. It has 5 different rotational horseshoes having (or not) heteroclinic connections. In the paper we will define sub-surfaces $S_i$ associated to the dynamics and a graph whose vertices are those sub-surfaces and the edges are given by the adjacency; some will be oriented by the relation of heteroclinic connections. We will see that a homeomorphism with big rotation set resembles a lot the one of this example.
• Figure 3: An example of a homeomorphism of a surface of genus 4 having 5 horseshoe classes $S_i$, and some horseshoes in these classes, with the graph $\mathcal{T}$ associated to it. The images of the rectangles are represented as thick lines, we suppose that all the pictured intersections are Markovian. In this case, the ergodic rotation set of $f$ is the union of 4 pieces that are of dimension 2 (their closures are convex sets) and the rotation set of $f$ is the union of 4 convex sets, each of dimension 4. The oriented graph $\mathcal{T}$ has no oriented loop but is not a tree.
• Figure 4: Example of a homeomorphism with big rotation set, the associates surfaces $S_i$ and the graph $\mathcal{T}$. Note that the piece $\rho_0$ of $\operatorname{rot}_{\mathrm{erg}}(f)$ associated to $S_0$ is equal to $\{0\}$ (as $S_0$ has no genus, we have $i_*(H_1(S_0, \mathbf{R})) = \{0\}$).
• Figure 5: The counterexamples of Example \ref{['ExFinal']}: construction of $f_n$ for $n=4$. The black dots represent contractible fixed points. Top: first part of the proof; bottom: second part of the proof. Note that the second example gives a counterexample of Problem 2 of pollicott (note that the set $\rho(f)$ of pollicott is the punctual rotation set): the rotation set of points has nonempty interior while the rotation vectors of periodic orbits are contained in the union of two planes.

Theorems & Definitions (46)

• Definition 1.1
• Definition 1.2: Homological rotation sets
• Lemma 1.3
• Definition 1.4
• Theorem 1
• Theorem 2
• Proposition 3
• Proposition 4
• Conjecture 1.5
• Proposition 5