Minimal Homeomorphisms on the Torus

• Minimal homeomorphisms on the torus are continuous maps where every orbit is dense, and they exhibit complex rotation sets and invariant decompositions.
• The classification of minimal sets outlines three types based on the nature of their complement components, providing insight into the dynamics and geometric structure.
• Analyses of obstructions, factor maps, and conjugacy reveal intricate dependencies on ergodic theory, group actions, and descriptive set-theoretic complexities.

A minimal homeomorphism on the torus $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$ is a homeomorphism $f:\mathbb{T}^2 \rightarrow \mathbb{T}^2$ such that for every $x \in \mathbb{T}^2$ the orbit $\{f^n(x): n \in \mathbb{Z}\}$ is dense in $\mathbb{T}^2$. The study of minimal homeomorphisms on $\mathbb{T}^2$ intersects ergodic theory, topological dynamics, and the theory of group actions. Classification, structural results, and explicit constructions reveal complex phenomena absent in the one-dimensional case, especially regarding rotation sets, invariant decompositions, obstructions to factorization, and the descriptive set-theoretic complexity of conjugacy.

1. Structural Definitions and Foundational Results

A homeomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ is isotopic to the identity if it lies in the identity component of $\operatorname{Homeo}(\mathbb{T}^2)$. Given a lift $F:\mathbb{R}^2\to\mathbb{R}^2$ of $f$, the rotation set is defined as:

$$\rho(F) = \bigcap_{m\geq 1} \overline{\bigcup_{n \geq m} \left\{ \frac{F^n(z) - z}{n} : z \in \mathbb{R}^2 \right\}} \subset \mathbb{R}^2$$

The rotation set of $F$ projects to $\rho(f)$ in $\mathbb{T}^2$. The set $\rho(f)$ is always compact and convex (Kwakkel, 2010).

A nonempty closed invariant set $\mathcal{M} \subset \mathbb{T}^2$ is minimal if $f(\mathcal{M}) = \mathcal{M}$ and $\mathcal{M}$ contains no proper nonempty closed invariant subset; equivalently, every orbit in $\mathcal{M}$ is dense in $\mathcal{M}$.

Minimality is closely tied to periodic point-freeness: $f$ is minimal if and only if every forward orbit is dense, which, for homeomorphisms isotopic to the identity, frequently entails the absence of periodic points.

2. Classification of Minimal Sets for Non-Resonant Homeomorphisms

For non-resonant torus homeomorphisms isotopic to the identity, i.e., those with rotation set $\rho(f) = \{(\alpha, \beta)\}$ and $1, \alpha, \beta$ rationally independent, the minimal set structure admits a complete classification (Kwakkel, 2010):

• If $\mathcal{M} \neq \mathbb{T}^2$, then the connected components $\Sigma_k$ of the complement $\mathbb{T}^2 \setminus \mathcal{M}$ are categorized as follows:
  1. Type I: All $\Sigma_k$ are open disks.
  2. Type II: Each $\Sigma_k$ is either an open disk or an essential annulus ($\pi_1(\Sigma_k) \to \pi_1(\mathbb{T}^2)$ injective for annuli), with at least one annulus present.
  3. Type III: $\mathcal{M}$ is an extension of a Cantor set. There exists a semi-conjugacy $\varphi$ homotopic to the identity mapping $f$ to another non-resonant homeomorphism whose minimal set is a Cantor set. The domains $\Sigma_k$ are interiors of filled-in components associated to this Cantor set.

In all cases above, if $\mathcal{M} \neq \mathbb{T}^2$, $f$ has a unique minimal set, and the orbit closure and nonwandering set coincide with $\mathcal{M}$ (cases I, II).

A summary of minimal set types:

Type	Complement Components	Minimal Set $\mathcal{M}$
I	All open disks	Cantor-like continuum (quasi-Sierpiński)
II	Mix of disks + essential annuli	Cantor $\times S^1$, extensions possible
III	Interiors filled-in from Cantor	Extension of Cantor, countable disk boundaries

3. Rotation Set Geometry and Pseudo-Rotations

Minimal torus homeomorphisms often fall into two regimes governed by the structure of the rotation set (Kocsard, 2016, Kwakkel, 2010):

• Pseudo-rotations: Rotation set is a singleton. Such $f$ may or may not be minimal, but when minimal, every orbit behaves uniformly.
• Non-singleton rotation sets: For minimal $f$ isotopic to identity, $\rho(\tilde f)$ must be either (i) a singleton (pseudo-rotation), or (ii) a segment of irrational slope with no rational points. Rational-slope segments are excluded by bounded deviation theorems; see (Kocsard, 2016).

Rotational deviations for lifts $\tilde f$ are measured relative to a direction $v$:

$$\Delta_v^{(n)}(z) = \langle \tilde f^n(z) - z, v \rangle - n\alpha$$

where uniformly bounded deviations are critical for factorization structures.

4. Obstructions, Factor Maps, and Extensions

Given a periodic point-free homeomorphism, two principal obstructions govern whether $f$ can admit an irrational circle rotation as a topological factor (Kocsard, 2019):

• Annularity: If $f^n$ restricts most orbits to homological bands, factorization is impossible.
• Large wandering domains: The existence of wandering sets with large or essential components precludes a global semi-conjugacy.

If these obstructions are absent and $f$ exhibits uniformly bounded rotational deviations (in a rational direction), $f$ admits a factor to an irrational circle rotation. This equivalence is formalized via the construction of a skew-product absorbing average rotation, and the formation of essential annular continua whose levels define the factor map.

Example constructions include suspended Denjoy homeomorphisms and product systems, in which one factor is a Denjoy minimal set.

5. Group Actions: Minimality and Algebraic Properties

Explicit constructions exist for finitely generated simple groups acting minimally on the torus (Hyde et al., 2021). For $n \geq 3$, one defines fast $n$-ring groups $G_n$ generated by homeomorphisms with cyclic support segments on $S^1$, and takes $H_n = [G_n, G_n]$.

Actions of $H_n$ on $\mathbb{T}^2$ are constructed via maps of the form:

$$(\theta,\phi) \mapsto (\theta \cdot x,\,\phi + \alpha\,\chi_{\operatorname{Supp}(x)}(\theta)) \pmod{1}$$

where $\alpha$ is irrational and $\chi_{\operatorname{Supp}(x)}$ is the support indicator. This yields faithful, minimal actions, providing the first examples of finitely generated simple left-orderable groups acting minimally on $\mathbb{T}^2$.

Algebraic properties:

• Infinite, simple, non-amenable, left-orderable
• Infinite commutator width (arising from unbounded quasimorphisms)
• Dense orbits under the group action; no nontrivial homomorphism to $\mathbb{Z}$.

Rotation sets generated by these maps form infinite-rank subgroups.

6. Conjugacy Classification and Descriptive Set Theory

Recent results establish a key contrast between circle and torus dynamics (Peng, 30 Dec 2025). While in the circle case minimal homeomorphisms can be classified via Poincaré’s rotation number, on $\mathbb{T}^2$ topological conjugacy of minimal homeomorphisms resists classification by countable structures.

An explicit anti-classification theorem: for the equivalence relation of topological conjugacy restricted to minimal homeomorphisms, there is no Borel reduction to any equivalence relation arising from countable structures. A Borel map $\alpha \mapsto f_\alpha$ from $[0,1]^\mathbb{N}$ to minimal homeomorphisms is constructed such that $f_\alpha$ and $f_\beta$ are conjugate if and only if $\alpha$ and $\beta$ are $c_0$-equivalent.

This result settles negatively Smale’s program of classification for minimal surface dynamics and illustrates the prevalence of turbulence in the orbit equivalence relations on $\mathbb{T}^2$.

7. Open Questions and Generalizations

Open directions include:

• Extension of factorization dichotomies and explicit classifications to higher genus surfaces and higher-dimensional tori.
• Rigidity and regularity results in smoother categories, specifically the classification of $C^\infty$ minimal diffeomorphisms and possible finer dynamical invariants.
• Interplay between ergodicity, unique ergodicity, and rotation set geometry for minimal systems with irrational slope segments.

Existing frameworks suggest that the combination of rotation set analysis, obstruction theory (annularity, wandering domains), and descriptive set-theoretic complexity will drive further research into minimal dynamics on compact surfaces and manifolds.

• Ferry Kwakkel, "Minimal sets of non-resonant torus homeomorphisms" (Kwakkel, 2010).
• Hyde, Lodha, Rivas, "Two new families of finitely generated simple groups of homeomorphisms of the real line" (Hyde et al., 2021).
• Anti-classification result for minimal toral homeomorphisms (Peng, 30 Dec 2025).
• Kocsard, "On the dynamics of minimal homeomorphisms of $\mathbb{T}^2$ which are not pseudo-rotations" (Kocsard, 2016).
• Kocsard, "Periodic point free homeomorphisms and irrational rotation factors" (Kocsard, 2019).