Ergodicity of (co)expanding on average random dynamical systems

Dmitry Dolgopyat; Jonathan DeWitt; Zhiyuan Zhang

arxiv: 2605.21199 · v1 · pith:VE4KXGC5new · submitted 2026-05-20 · 🧮 math.DS

Ergodicity of (co)expanding on average random dynamical systems

Jonathan DeWitt , Dmitry Dolgopyat , Zhiyuan Zhang This is my paper

Pith reviewed 2026-05-21 01:29 UTC · model grok-4.3

classification 🧮 math.DS

keywords ergodicityrandom dynamical systemsstable ergodicityexpansion on averagerotations on spherespectral gapstatistical limit theoremsLyapunov exponents

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The pith

Two rotations generating a dense subgroup make random dynamics on the sphere stably ergodic for all dimensions at least 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves ergodicity for random dynamical systems that expand on average and satisfy irreducibility conditions. It applies the result to show that if two elements of SO(d+1) generate a dense subgroup for d at least 2, then their random iterations on the sphere S^d are stably ergodic, extending prior knowledge limited to even dimensions. The argument yields a spectral gap for the transfer operator along with statistical limit theorems, and it remains valid even when some Lyapunov exponents are zero. A sympathetic reader would care because the result supplies concrete mixing properties for random rotation systems that arise in geometry and physics.

Core claim

Under expansion-on-average and irreducibility conditions the random dynamical system is ergodic. In particular, when R1 and R2 in SO(d+1) with d greater than or equal to 2 generate a dense subgroup, the random dynamics they induce on S^d is stably ergodic; this implies a spectral gap and statistical limit theorems even in the presence of zero Lyapunov exponents.

What carries the argument

Expansion-on-average condition together with irreducibility, which together force orbits to spread and preclude nontrivial invariant measures.

If this is right

The associated transfer operator possesses a spectral gap.
Central limit theorems and other statistical limit theorems hold for suitable observables.
Ergodicity persists under small perturbations of the generating rotations.
The unique stationary measure is equivalent to the volume measure on the sphere.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same expansion-on-average technique could apply to random actions of other compact Lie groups on homogeneous spaces.
Numerical checks of dense subgroups generated by two rotations in SO(3) would provide concrete illustrations of the mixing rates.
The result suggests that many random walks on manifolds with neutral directions still admit statistical predictability.

Load-bearing premise

The random maps must satisfy the stated expansion-on-average and irreducibility conditions.

What would settle it

An explicit pair of matrices in SO(3) that generate a dense subgroup yet leave a nontrivial measurable set invariant under the random iteration on S^2 would falsify the claim.

read the original abstract

We prove ergodicity for random dynamics satisfying some expansion and irreducibility conditions. As a particular application, we show that if $R_1,R_2\in \mathrm{SO}(d+1)$, $d\ge 2$, generate a dense subgroup, then the random dynamics of $R_1$ and $R_2$ on $S^d$ is stably ergodic. Previously this was only known to hold in even dimensions. As a consequence, we deduce spectral gap and statistical limit theorems for such systems. In particular, our results apply in the presence of zero Lyapunov exponents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a general ergodicity criterion for random systems that expand only on average plus irreducibility, then applies it to extend stable ergodicity for dense random rotations on the sphere to odd dimensions and zero Lyapunov exponents.

read the letter

This paper proves ergodicity for random dynamical systems under expansion-on-average and irreducibility conditions, then applies the result to show that if two elements of SO(d+1) generate a dense subgroup, their random dynamics on the sphere S^d is stably ergodic for all d greater than or equal to 2. This includes odd dimensions and situations with zero Lyapunov exponents, leading to spectral gap and limit theorems. Previously the sphere result was known only in even dimensions. The extension to odd dimensions is the clear novelty here. The general criterion looks like it could be applied to other random systems where uniform expansion fails but average expansion holds. The work handles the zero-exponent case in the consequences, which is a plus since that often blocks spectral gap results. The soft spot is in confirming that dense generation implies the required expansion-on-average condition. It is not obvious that the averaged expansion is strict when some exponents vanish, and the argument needs to be checked for whether it works uniformly in all dimensions without hidden even-dimensional assumptions. If that step is fully rigorous and explicit, the claims hold up; otherwise it is the place where a referee might ask for more detail or a counterexample check. Readers in ergodic theory and random dynamical systems will get the most from this, especially those studying actions on spheres or homogeneous spaces. It is the kind of paper that advances the toolkit for proving ergodicity in random settings. I think this deserves a serious referee. The combination of a general theorem and a new application in an area with prior results makes it worth the review time. Recommendation: send to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves ergodicity for random dynamical systems satisfying expansion-on-average and irreducibility conditions. As an application, it shows that if two rotations R1, R2 in SO(d+1) with d ≥ 2 generate a dense subgroup, then the random dynamics on S^d is stably ergodic; this extends prior results to odd dimensions and holds in the presence of zero Lyapunov exponents, yielding spectral gap and statistical limit theorems as consequences.

Significance. If the central claims hold, the work supplies a general criterion for ergodicity in random dynamical systems that tolerates neutral directions, which is a notable advance over results requiring uniform expansion. The resolution of the odd-dimensional sphere case for dense rotation pairs is a concrete contribution to the literature on random maps on manifolds, with downstream implications for mixing and limit theorems.

major comments (1)

[§5] §5 (Application to rotations on spheres): The argument that dense generation of a subgroup in SO(d+1) implies the expansion-on-average condition (Definition 2.1 or 2.3) is invoked without an explicit, dimension-independent verification that the averaged log-norm of the derivative is strictly positive (rather than merely non-negative) when some Lyapunov exponents vanish. This step is load-bearing for the stable ergodicity claim in odd dimensions and must be supplied or referenced with a concrete estimate.

minor comments (2)

[§2] Notation for the averaged expansion quantity could be made more uniform between the general theorem and the application to avoid reader confusion.
[§5] A short remark on how the irreducibility condition is checked for the rotation example would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below and will make the corresponding revisions to strengthen the exposition in Section 5.

read point-by-point responses

Referee: [§5] §5 (Application to rotations on spheres): The argument that dense generation of a subgroup in SO(d+1) implies the expansion-on-average condition (Definition 2.1 or 2.3) is invoked without an explicit, dimension-independent verification that the averaged log-norm of the derivative is strictly positive (rather than merely non-negative) when some Lyapunov exponents vanish. This step is load-bearing for the stable ergodicity claim in odd dimensions and must be supplied or referenced with a concrete estimate.

Authors: We agree that the passage from density of the subgroup generated by R1 and R2 to the strict positivity of the averaged log-norm of the derivative (rather than mere non-negativity) deserves a more explicit, dimension-independent argument, especially when zero Lyapunov exponents are present. In the current manuscript this implication is obtained from the combination of density, the irreducibility condition, and the geometry of the sphere, but the estimate is not written out in full detail. We will add a short self-contained lemma in the revised Section 5 that supplies a concrete lower bound: using the density of the subgroup in SO(d+1) for odd d, we show that there exists ε>0 (independent of the particular dense pair) such that the average of log‖DR_i(x)‖ over the stationary measure is at least ε>0. The argument relies on the fact that any invariant neutral subbundle would contradict density for odd-dimensional spheres and on a uniform control of the derivative norms away from the neutral directions. This addition will be purely expository and will not change the statements or proofs of the main theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a direct proof of ergodicity from the stated expansion-on-average and irreducibility conditions, with the application consisting of a separate verification that dense generation in SO(d+1) implies those conditions (including for zero Lyapunov exponents). No quoted equations or steps reduce the conclusion to a fitted input, self-definition, or load-bearing self-citation chain; the central argument remains independent of the target ergodicity statement and is presented as a standard mathematical implication rather than a renaming or ansatz smuggling. This is the expected non-finding for a theorem-proof structure with externally verifiable hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background facts from ergodic theory and Lie-group geometry rather than new fitted parameters or postulated entities.

axioms (2)

standard math Standard properties of the special orthogonal group SO(d+1) and the notion of dense subgroups
Used to formulate the concrete application on the sphere.
domain assumption Ergodicity follows from expansion-on-average together with irreducibility for random dynamical systems
This is the load-bearing hypothesis of the general theorem stated in the abstract.

pith-pipeline@v0.9.0 · 5627 in / 1498 out tokens · 39131 ms · 2026-05-21T01:29:08.114774+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

On the ergodicity of partially hyperbolic systems

arXiv: 2502.14042 [math.DS]. [BW10] Keith Burns and Amie Wilkinson. “On the ergodicity of partially hyperbolic systems”. Ann. of Math. (2) 171.1 (2010), pp. 451–489. doi: 10.4007/annals.2010.171.451. [CFS24] Michele Caselli, Enric Florit-Simon, and Joaquim Serra. “Fractional Sobolev spaces on Riemannian manifolds”. Math. Ann. 390.4 (2024), pp. 6249–6314. ...

work page doi:10.4007/annals.2010.171.451 2010
[2]

Simultaneous linearization of diffeomorphisms of isotropic mani- folds

doi: 10.1016/j.aim.2022.108625. [CM06] Nikolai Chernov and Roberto Markarian. Chaotic billiards. Vol. 127. Mathematical Sur- veys and Monographs. American Mathematical Society, Providence, RI, 2006, pp. xii+316. doi: 10.1090/surv/127. REFERENCES 27 [DeW24] Jonathan DeWitt. “Simultaneous linearization of diffeomorphisms of isotropic mani- folds”. J. Eur. M...

work page doi:10.1016/j.aim.2022.108625 2022

[1] [1]

On the ergodicity of partially hyperbolic systems

arXiv: 2502.14042 [math.DS]. [BW10] Keith Burns and Amie Wilkinson. “On the ergodicity of partially hyperbolic systems”. Ann. of Math. (2) 171.1 (2010), pp. 451–489. doi: 10.4007/annals.2010.171.451. [CFS24] Michele Caselli, Enric Florit-Simon, and Joaquim Serra. “Fractional Sobolev spaces on Riemannian manifolds”. Math. Ann. 390.4 (2024), pp. 6249–6314. ...

work page doi:10.4007/annals.2010.171.451 2010

[2] [2]

Simultaneous linearization of diffeomorphisms of isotropic mani- folds

doi: 10.1016/j.aim.2022.108625. [CM06] Nikolai Chernov and Roberto Markarian. Chaotic billiards. Vol. 127. Mathematical Sur- veys and Monographs. American Mathematical Society, Providence, RI, 2006, pp. xii+316. doi: 10.1090/surv/127. REFERENCES 27 [DeW24] Jonathan DeWitt. “Simultaneous linearization of diffeomorphisms of isotropic mani- folds”. J. Eur. M...

work page doi:10.1016/j.aim.2022.108625 2022