Rigidity on the 2-Torus: Sarnak’s Conjecture for a Big Class of Maps

Rigidity on the 2-Torus: Sarnak’s Conjecture for a Big Class of Maps

Introduction: when a “two-dimensional rotation” acts almost like the identity

If you’ve ever played with a system that “looks like a rotation,” you’ve probably noticed a frustrating truth: on a circle, life is much easier than on a torus. On the circle, Poincaré’s rotation number basically tells you what the system is, up to a soft notion of equivalence. But on the two-torus, even if everything shares the same average rotation behavior, the dynamics can still be wildly different.

That’s exactly the tension addressed in new research from Chang–Wang–Zhou (arXiv:2608.26976). Their focus is quantitative rigidity for pseudo-rotations on (\mathbb T^2): maps that behave like an irrational rotation “on average,” but aren’t necessarily linear. The headline is a clean kind of control: under a specific ((C,\delta))-deviation condition (a quantitative strengthening of “bounded mean motion”), the authors prove strong statements like:

• certain Hölder continuous, area-preserving pseudo-rotations are (C^0)-rigid along a subsequence, often with exponential decay;
• other smooth-ish pseudo-rotations with “strong non-Brjuno” arithmetic behavior are more rigid at lower derivative levels with superpolynomial decay;
• and skew products over circle rotations also become rigid—under the right arithmetic hypotheses—leading to Sarnak’s conjecture for all these classes.

So yes: this is deep dynamics, but the outcome is very concrete. The paper essentially builds a pipeline:
orbit deviation bounds → a “free-disk estimate” → explicit proximity to the identity → a Möbius-disjointness statement (Sarnak).

Why This Matters: rigidity isn’t just pretty—it’s a practical guarantee

Rigidity results like these are significant right now because “zero entropy” systems are increasingly used as testbeds and assumptions in data-driven dynamical modeling, especially when you want to avoid chaotic blow-ups but still capture long-term structure. Sarnak’s conjecture is one of the main ways mathematicians certify that such structured systems do not “accidentally correlate” with the Möbius function.

The practical hook is this: Sarnak’s conjecture is a rigorous form of “no hidden arithmetic resonance.” If your system is topologically tame (entropy zero) and rigid enough in the (C^0) sense, you can rule out subtle number-theoretic patterns. That’s exactly the kind of guarantee you want when using deterministic dynamics as a generator of signals or as an emulator of “structured noise” in applications.

A concrete today scenario: imagine you’re testing a deterministic simulator for long-range dependence while also checking it doesn’t encode spurious number-theoretic fingerprints. Sarnak’s conjecture provides a blueprint for what “fingerprint-free” means: for continuous observables, orbit averages weighted by Möbius go to zero. The framework in this paper expands the class of deterministic torus systems for which that guarantee holds.

And importantly, this paper doesn’t just reprove known cases—it pushes a quantitative route. Previous rigidity-to-Sarnak strategies often relied on more geometric regularity or earlier criteria. Here, the authors’ ((C,\delta))-deviation condition is designed to be checkable (at least in principle) and it plugs into a new quantitative disk estimate. That’s a methodological upgrade: they make orbit deviation control the geometry of the map in a way that’s explicit enough to drive decay rates.

Finally, relative to “AI research”: you can think of this as a kind of theorem about stability of long-term behavior under repeated composition. In machine learning terms, it’s about when a complex transformation, when iterated, returns close to the “reference transformation” (the identity) at a controlled rate. That’s conceptually similar to how algorithms sometimes need uniform contraction/recurrence guarantees to avoid pathological drift. Of course, the objects here are different—but the underlying theme is the same: you don’t want your iterative system to wander unpredictably forever.

Main Content Sections

From rotation sets to pseudo-rotations: the torus version of “what rotation means”

On the circle, an orientation-preserving homeomorphism (f) with irrational rotation number is essentially an irrational rotation, up to semi-conjugacy. On (\mathbb T^2), there is no single rotation number. Instead, you talk about the rotation set, built from how the lifted dynamics drifts on average.

Here’s the intuitive picture the paper uses:

• Lift your torus map (f:\mathbb T^2\to\mathbb T^2) to (\tilde f:\mathbb R^2\to\mathbb R^2).
• Track displacement (\tilde f^n(\tilde z)-\tilde z) over (n) steps.
• Divide by (n), and see if the limit exists.

If that limiting average displacement is a single vector independent of the starting point, the map is called a pseudo-rotation. If the vector is irrational, the system has “irrational drift” but still no periodic obstruction.

Why pseudo-rotations can still be weird: even with a single rotation vector, the map can have weak mixing behavior—so rotation data alone doesn’t fully determine the dynamics on (\mathbb T^2).

That’s why the paper’s “extra assumption” is crucial: it adds a quantitative control mechanism (the deviation condition) to rule out those pathological behaviors.

Rotation vectors and the role of lifts

A rotation vector (\omega) for a pseudo-rotation is defined so that (for a suitable lift) the displacement averages converge:
[
\frac{\tilde f^n(\tilde z)-\tilde z}{n}\to \vec\omega.
]
They then consider (\omega) modulo (\mathbb Z^2), so (\bar\omega\in \mathbb R^2/\mathbb Z^2) is the “true” rotation vector on the torus.

The ((C,\delta))-deviation condition: turning “almost bounded motion” into a rate

The paper builds a quantitative deviation notion that strengthens the older bounded mean motion condition.

• Bounded mean motion corresponds to a deviation rate like (O(n^{-1})), i.e. ((C,0))-deviation in their framework.
• Their new condition is ((C,\delta))-deviation:
  [
  \Delta_N(f)\le C\,N^{\delta-1}\quad\text{for all }N\in\mathbb N,
  ]
  with (\delta\in[0,1)).

What is (\Delta_N(f)) conceptually? It measures how far the (n)-step displacement deviates from the expected linear drift by (\omega). The key point is: the deviation doesn’t just go to zero—it goes to zero at a controlled power rate (or faster when (\delta) is smaller).

Here’s the comparison in a compact way:

This isn’t just cosmetic. The paper repeatedly leverages the deviation exponent (\delta) to get explicit control on geometry (free disks) and then to translate that into rigidity decay rates.

Also worth noting: they explicitly say their generalization method doesn’t extend to the previously studied “parallel bounded deviation” case inside this ((C,\delta)) framework—so this result is “targeted,” not a universal blanket.

The free-disk estimate: the clever bridge from orbit deviation to rigidity

This is the core technical ingredient—and it’s worth appreciating even if you don’t chase the full proof.

On a torus, to show a map (f) is close to the identity in (C^0), it helps to understand whether the map can “fit” a region into itself without overlapping. That leads to free disks:

• A free disk (D\subset \mathbb T^2) is an embedded topological disk such that
  [
  f(D)\cap D=\varnothing.
  ]

If a map is close to the identity, it can’t easily move every disk away from itself. Conversely, if there are big free disks, the map must be moving points significantly.

So the strategy becomes:

1. Use the deviation condition to control the maximum area/size of free disks.
2. Convert the existence of large free disks into a lower bound on displacement.
3. Invert that: if free disks can’t be large, the map must be close to identity.

The paper’s estimate (proved through a combination of Brouwer-type arguments and area-preserving dynamics) gives something like:

• under ((C,\delta))-deviation, the area of a free disk is bounded in terms of the rotation vector size (\omega) and parameters ((C,\delta)).

Once you have “free disks are small,” Hölder (or smoother) regularity lets you translate that into a global (C^0) bound for (f^n) near the identity.

This is exactly the promised pipeline from the abstract, and the paper highlights that the novelty is the quantitative version of the disk estimate.

Rigidity results for pseudo-rotations: exponential, superpolynomial, and derivative drop

Now we get to the main theorems—the “what happens if you assume deviation?” part.

The exponential-rate (C^0)-rigidity for Hölder super-Liouville pseudo-rotations

Consider an area-preserving pseudo-rotation (f) on (\mathbb T^2) that is:

• irrational,
• Hölder continuous with exponent (a\in(0,1]),
• with rotation vector satisfying a strong super-Liouville condition,
• and satisfying ((C,\delta))-deviation with (\delta\in[0,1/2)).

Then the paper proves that (f) is (C^0)-rigid with an exponential decay rate. In their language: there exists a subsequence (nj) such that
[
d{C^0}(f^{nj},\mathrm{id}) \le C1 e^{-C2 nj}.
]

This is where number theory meets dynamics. Super-Liouville behavior means there are very good rational approximations along subsequences; in turn, that forces the rotation drift (\|n_j\omega\|) to be extremely small, which feeds into the free-disk bound and ultimately into rigidity.

Semi-irrational pseudo-rotations: rigidity of order (C^{k-1}) with superpolynomial decay

If the rotation vector is semi-irrational—meaning there is a nontrivial integer relation (c\omega1+d\omega2+e=0)—the arithmetic constraints can be weakened from “strong super-Liouville” to a “strong non-Brjuno” type condition.

For (C^k) pseudo-rotations (with (2\le k<\infty)) that are semi-irrational, they show:

• under ((C,\delta))-deviation with (\delta\in[0,1/2)),
• the map is (C^{k-1})-rigid with superpolynomial decay.

So unlike the purely (C^0) story, higher derivatives don’t stay uniformly controlled under the same argument. The result drops by one derivative order (from (k) to (k-1)), which is typical in many rigidity-to-regularity transfer arguments.

What this means in plain terms

If you’re picturing the torus as a doughnut surface and the pseudo-rotation as a “rotation-like drift,” rigidity means:

• after many steps, the system comes back extremely close to where it started,
• and the return speed can be quantified.

The deviation condition prevents “spreading free regions too widely,” which is exactly what would stop the iterates from returning close to identity.

Skew products over irrational rotations: when extensions still behave rigidly

The third rigidity class is about skew products
[
T_{\alpha,h}(x,y)=(x+\alpha,\; y+h(x)).
]
Here (\alpha) is irrational and (h) is Hölder continuous. These are natural because they’re “rotation in the base + a vertical cocycle shift.”

A few key constraints matter:

• The map must be isotopic to identity, which becomes (\deg(h)=0).
• The skew product is a pseudo-rotation under the right lift/displacement conditions, so rotation-vector tools apply.

Then the rigidity depends not only on ((C,\delta))-deviation but also on an arithmetic threshold involving the irrationality measure (\mu(\alpha)). Recall:
- Diophantine numbers have finite irrationality measure.
- Liouville numbers have infinite irrationality measure.
- The theorem requires (\mu(\alpha)) be sufficiently large (they formulate it with explicit inequalities depending on (a) and (\delta)).

Under those hypotheses, they prove:

• (T_{\alpha,h}) is (C^0)-rigid with polynomial decay rate along a subsequence.

Why the arithmetic condition can’t just be removed

The paper also discusses sharpness limits using prior work (like De Faveri) showing there are skew products where rigidity along convergent denominators fails to have polynomial decay—even when the skew product is Hölder (even (C^1) in some constructions). That’s why their ((C,\delta))-deviation assumption and the irrationality threshold on (\alpha) are doing real work.

Consequence: these systems satisfy Sarnak’s conjecture

Now for the punchline: rigidity implies Möbius disjointness.

Sarnak’s conjecture (roughly) says: for any topological dynamical system with zero entropy, the Möbius-weighted averages of observables along orbits go to zero.

The authors leverage a criterion due to Kanigowski–Lemańczyk–Radziwiłł (the “KBSZ criterion” in the paper) plus a key fact:

• (C^0)-rigidity (with polynomial decay) implies zero topological entropy and strong enough regularity in orbit averages to conclude Sarnak’s conjecture.

They explicitly state that the systems in their three rigidity categories—(1) the Hölder strong super-Liouville pseudo-rotations, (2) the semi-irrational strong non-Brjuno pseudo-rotations (with derivative rigidity), and (3) the Hölder skew products under large irrationality measure—satisfy Sarnak’s conjecture.

So the narrative becomes:

1. Assume ((C,\delta))-deviation (and suitable Hölder + arithmetic).
2. Prove quantitative rigidity via free-disk bounds.
3. Use rigidity decay to verify the criterion’s prerequisites.
4. Conclude Sarnak.

It’s satisfying because the dynamics aren’t being forced into a linear model; instead, the system is shown to repeatedly return close to identity at a measurable rate, which is exactly the kind of “tame arithmetic behavior” Sarnak’s conjecture is about.

Key Takeaways

• New quantitative condition: The paper introduces and uses ((C,\delta))-deviation, a rate-based strengthening of bounded mean motion.
• Main technical bridge: A quantitative free-disk estimate converts orbit deviation bounds into explicit bounds on the size of free disks, which then control how close iterates get to the identity.
• Exponential rigidity: Hölder continuous, area-preserving irrational pseudo-rotations with strong super-Liouville rotation vectors and ((C,\delta))-deviation (with (\delta<1/2)) are (C^0)-rigid with exponential decay.
• Derivative-level rigidity drop: For (C^k) semi-irrational pseudo-rotations of strong non-Brjuno type, they get (C^{k-1})-rigidity with superpolynomial decay under the same deviation regime.
• Skew products also rigid (polynomially): Hölder skew products (T_{\alpha,h}) with (\deg(h)=0) and suitable large irrationality measure (\mu(\alpha)), plus ((C,\delta))-deviation, are (C^0)-rigid with polynomial decay.
• Sarnak’s conjecture holds for all these systems: Using rigidity criteria (KBSZ-type arguments), the authors conclude Möbius disjointness for these classes, thanks to the zero-entropy and rigidity properties.
• Sharpness is not guaranteed beyond assumptions: The paper discusses constructions showing that without the deviation condition (or without the arithmetic thresholds), polynomial (C^0)-rigidity along natural subsequences can fail.

If you want, I can also rewrite the main results as an “if you check these 3 conditions, you automatically get rigidity + Sarnak” checklist, tailored to the exact class (pseudo-rotation vs semi-irrational vs skew products).