The Le Calvez–Yoccoz cyclic-order invariant at a locally maximal fixed point: structure, completeness for germ conjugacy, extension, and computation

In 1997 Le Calvez and Yoccoz proved an index theorem for an orientation-preserving homeomorphism of the plane at a fixed point that is a locally maximal invariant set and neither a sink nor a source: the sequence of fixed point indices of the iterates is periodic, equal to 1 off multiples of a period q and to 1−rq on multiples, and from this they deduced that no finitely punctured two-sphere (in particular no open annulus) admits a minimal homeomorphism. Within the same work they constructed a conjugacy invariant of the germ of such a homeomorphism: a compact, separable, cyclically ordered set $J_+^f$ — together with its dual forming the full invariant $\mathcal J^f$ — defined as the projective limit of finite cyclically ordered sets attached to a filtering family of Jordan domains isolating the fixed point, and equipped with a cyclic-order-preserving automorphism $\widehat f$ whose finite invariant part has cardinality rq and rational rotation p/q. The invariant was never developed further. The open problem of what the invariant sees about the local dynamics is stated here as four linked questions: characterize the elements of $J_+^f$ representable by stable/unstable branches and the structure of the non-representable part; decide whether the invariant is complete for orientation-preserving conjugacy of germs; extend the construction to fixed points that are not locally maximal; and compute the invariant in explicit examples beyond radially symmetric models. No work addressing any of the four questions is known in the literature; all four remain open.

Background

Let $f:W\to W'$ be an orientation-preserving homeomorphism between open neighborhoods of the origin in the plane with $f(0)=0$ and $0$ an isolated fixed point. The (Lefschetz, or Poincaré) index $i(0,f)$ of $f$ at $0$ is the degree of the map $x\mapsto(f(x)-x)/\|f(x)-x\|$ restricted to a small circle around $0$; it depends only on the conjugacy class of the germ of $f$ at $0$ (the germ is the equivalence class of $f$ under restriction to smaller neighborhoods; conjugacy means composition with a homeomorphism fixing $0$). A classical theme is the sequence of indices of the iterates, $i(0,f^k)$, $k\ge1$: it is constrained by arithmetic congruences, equals $1$ for every $k$ when $0$ is accumulated by periodic orbits, and before 1997 its behavior in the purely topological category was understood mainly for differentiable germs, where linearization techniques apply, and in a few special situations.

Le Calvez and Yoccoz settled the following topological case. Assume that $0$ is neither a sink nor a source — for every sufficiently small neighborhood $V$ of $0$, neither $f(V)\subset V$ nor $V\subset f(V)$ — and that $0$ is a locally maximal invariant set, meaning that some neighborhood $V$ of $0$ satisfies $\bigcap_{k\in\mathbb Z}f^{-k}(\overline V)=\{0\}$: the only full orbit that stays in $V$ for all positive and negative time is the fixed point itself. Then there exist integers $q\ge1$ and $r\ge1$ such that $i(0,f^k)=1$ whenever $q$ does not divide $k$, and $i(0,f^k)=1-rq\le0$ whenever $q$ divides $k$. From this they deduced that no homeomorphism of the two-sphere punctured at a finite set can be minimal (a homeomorphism is minimal if every one of its orbits is dense), completing the earlier work of Handel for the sphere with at least three punctures; in particular the open annulus $\mathbb R/\mathbb Z\times\mathbb R$ admits no minimal homeomorphism.

The proof constructs, beyond the numerical index, a finer object which is the subject of the present problem. For a Jordan domain $U$ (an open topological disk bounded by a simple closed curve) belonging to a certain filtering family of nested domains that isolate $0$, let $\Delta_+^{(U)}$ be the invariant continuum of points of $\overline U$ whose forward orbits remain in nested such domains, and $\Delta_-^{(U)}$ the analogous object for backward orbits. The set $J_+^{(U)}$ of connected components of $\overline U\setminus\Delta_+^{(U)}$ that meet $\Delta_-^{(U)}$ is a finite cyclically ordered set, the cyclic order (a combinatorial betweenness relation for objects arranged around a circle) being induced by the orientation of the boundary of $U$. For nested domains $V\subset U$ there are natural surjective bonding maps $j_{V,U}:J_+^{(V)}\to J_+^{(U)}$ that preserve the cyclic order. The invariant is the projective (inverse) limit

$$J_+^f=\varprojlim_{U}J_+^{(U)},$$

a compact separable cyclically ordered set carrying an order-preserving automorphism $\widehat f$ induced by $f$; together with the dual construction this forms the full invariant $\mathcal J^f$. The invariant contains a finite $\widehat f$-invariant subset of exactly $rq$ elements on which $\widehat f$ acts as a rotation by $p/q$ of a turn, which yields a rational rotation number for the germ. Crucially, the invariant is a conjugacy invariant: if the germs of $f$ and $g$ are conjugate by an orientation-preserving homeomorphism, then $J_+^f$ and $J_+^g$ are isomorphic as cyclically ordered sets via a map conjugating $\widehat f$ to $\widehat g$. The invariant therefore refines the entire index sequence, since the pair $(q,r)$, and hence every value $i(0,f^k)$, is read off from it.

At the finite levels, the elements of $J_+^{(U)}$ correspond to pieces of the stable and unstable sets of $0$ — the branches of the local dynamics, that is, the arc-components of the sets of points whose forward, respectively backward, orbits converge to $0$. However, the 1997 paper establishes only the basic properties of the invariant (compactness, separability, conjugacy invariance, the finite rotation part) and leaves its theory undeveloped: the relationship between arbitrary elements of the projective limit, the branch structure, and the dynamics is not worked out; the construction uses the locally maximal hypothesis in an essential way and is not available for more general fixed points; and no computation of the invariant in explicit examples beyond elementary radially symmetric models is given. The problem below asks what this invariant actually sees about the local dynamics.

Problem Statement

Determine what the Le Calvez–Yoccoz invariant sees about the local dynamics of an orientation-preserving homeomorphism of the plane at a fixed point that is a locally maximal invariant set and neither a sink nor a source. The common objective uniting the requested results is to establish to what extent the cyclically ordered invariant $(\mathcal J^f,\widehat f)$ — equivalently its component $J_+^f$, the projective limit of the finite cyclically ordered sets $J_+^{(U)}$ over the filtering family of Jordan domains isolating the fixed point — characterizes the conjugacy class and the dynamics of the germ of $f$ at $0$, and how far the construction reaches. Specifically:

1. Structure. Characterize the elements of $J_+^f$ that are representable by stable/unstable branches of $f$ at $0$ (limits of finite-level components that correspond to actual branch pieces of the local dynamics), and describe the structure of the set of non-representable elements — in particular whether it can be nonempty, and what it measures dynamically.

2. Completeness. Decide whether $(\mathcal J^f,\widehat f)$ is a complete invariant for orientation-preserving conjugacy of germs: if the invariants of two germs satisfying the hypotheses are isomorphic as cyclically ordered sets equipped with their automorphisms, must the germs be conjugate by an orientation-preserving homeomorphism? If not, exhibit non-conjugate germs with isomorphic invariants and identify the additional data that would complete the invariant.

3. Extension. Extend the construction of the invariant — preserving conjugacy invariance and the compatibility with the index sequence — to isolated fixed points that are not locally maximal invariant sets, that is, to isolated fixed points for which local maximality fails (for example isolated fixed points that are neither sinks nor sources but are accumulated by periodic orbits, or saddle-type fixed points that are not locally maximal); the listed types belong to the target class only when local maximality fails, since a locally maximal fixed point that is neither a sink nor a source already satisfies the 1997 hypotheses and carries their invariant.

4. Computation. Compute the invariant explicitly for nontrivial classes of examples beyond radially symmetric models, and relate the computed invariants to other local invariants such as the index sequence, the rotation number, and the local rotation behavior.

References

1. Patrice Le Calvez, Jean-Christophe Yoccoz, 'Un théorème d'indice pour les homéomorphismes du plan au voisinage d'un point fixe' (An index theorem for the homeomorphisms of the plane near a fixed point), Annals of Mathematics (2) 146(2) (1997) 241–293. DOI: 10.2307/2952463, https://doi.org/10.2307/2952463.
2. Patrice Le Calvez, 'Dynamique des homéomorphismes du plan au voisinage d'un point fixe', Annales Scientifiques de l'École Normale Supérieure (4) 36(1) (2003) 139–171. DOI: 10.1016/S0012-9593(03)00005-3, https://doi.org/10.1016/S0012-9593(03)00005-3; open-access full text at https://www.numdam.org/item/ASENS_2003_4_36_1_139_0/.
3. Patrice Le Calvez, 'Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes ?', Annales Scientifiques de l'École Normale Supérieure (4) 41(1) (2008) 141–176. DOI: 10.24033/ASENS.2065, https://doi.org/10.24033/ASENS.2065; open-access full text at https://www.numdam.org/item/ASENS_2008_4_41_1_141_0/.
4. Michael Handel, 'There are no minimal homeomorphisms of the multipunctured plane', Ergodic Theory and Dynamical Systems 12(1) (1992) 75–83. DOI: 10.1017/S0143385700006593, https://doi.org/10.1017/S0143385700006593.
5. Patrice Le Calvez, Francisco R. Ruiz del Portal, José M. Salazar, 'Indices of the iterates of R^3-homeomorphisms at fixed points which are isolated invariant sets', Journal of the London Mathematical Society (2) 82(3) (2010) 683–696. DOI: 10.1112/jlms/jdq050, https://doi.org/10.1112/jlms/jdq050; preprint arXiv:0905.0088, https://arxiv.org/abs/0905.0088.
6. Frédéric Le Roux, 'Homéomorphismes de surfaces : théorèmes de la fleur de Leau-Fatou et de la variété stable' (Dynamics of surface homeomorphisms, topological versions of the Leau–Fatou flower theorem and the stable manifold theorem), Astérisque 292, Société Mathématique de France (2004), 120 p. ISBN: 2-85629-153-8, DOI: 10.24033/ast.619, https://doi.org/10.24033/ast.619; zbMATH 2116158, https://zbmath.org/2116158; open-access full text at https://www.numdam.org/item/AST_2004__292__1_0/; preprint arXiv:math/0210344, https://arxiv.org/abs/math/0210344.
7. Frédéric Le Roux, 'A topological characterization of holomorphic parabolic germs in the plane', Fundamenta Mathematicae 198(1) (2008) 77–94. DOI: 10.4064/fm198-1-4, https://doi.org/10.4064/fm198-1-4; preprint arXiv:0709.1398, https://arxiv.org/abs/0709.1398.
8. Frédéric Le Roux, 'Un indice qui affine l'indice de Poincaré-Lefschetz pour les homeomorphismes de surfaces' (An index that refines the Poincaré–Lefschetz index for surface homeomorphisms), Annals of Mathematics (2) 171(3) (2010) 1531–1589. DOI: 10.4007/annals.2010.171.1531, https://doi.org/10.4007/annals.2010.171.1531.
9. Jonathan Conejeros, 'The local rotation set is an interval', Ergodic Theory and Dynamical Systems 38(7) (2017) 2571–2617. DOI: 10.1017/etds.2016.129, https://doi.org/10.1017/etds.2016.129; preprint arXiv:1508.02152, https://arxiv.org/abs/1508.02152.
10. Christian Pries, 'A Remark to the Theorem of Le Calvez and Yoccoz', arXiv:1605.08873 (2016, revised 2019), https://arxiv.org/abs/1605.08873.