Counterexamples to double recurrence for non-commuting deterministic transformations

Shrey Sanadhya; Zemer Kosloff

REVIEW 3 major objections 6 minor 23 references

Counterexamples to double recurrence for non-commuting deterministic transformations

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For injective integer polynomials that vanish at the origin and have equal degree (both linear or both degree at least two), there exist mixing, zero-entropy, non-commuting transformations $T$ and $S$ and a positive-measure set $A$ whose…

desk verdict This paper likely closes the Frantzikinakis–Host double-recurrence question for non-commuting zero-entropy systems, and the main construction survives a close check; the two gaps I found are fixable without changing the statements. read the letter →

arxiv 2507.15528 v1 pith:5B5XG3KC submitted 2025-07-21 math.DS

Zemer Kosloff , Shrey Sanadhya This is my paper · ORCID

classification math.DS MSC 28D0537A0537A5037A3060F0560G1060G15

keywords MultiplerecurrenceNon-commutingtransformationsZeroentropyLocalcentrallimittheoremPolynomialiteratesMixingGaussianautomorphismsFourierdecay

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that double recurrence can fail completely for non-commuting, mixing, zero-entropy transformations whenever the two polynomial iterates have the same degree: both linear, or both of degree two or higher. For any injective integer polynomials $p_1,p_2$ with $p_1(0)=p_2(0)=0$, the authors construct transformations $T,S$ and a set $A$ of positive measure such that $A \cap T^{-p_1(n)}A \cap S^{-p_2(n)}A = \varnothing$ for every $n \in \mathbb{N}$. Together with the positive result already known for one linear and one higher-degree iterate, this settles the motivating question completely and marks the mixed-degree pair as the only case where double recurrence is guaranteed. The higher-degree construction uses a skew product over a zero-entropy base with a two-dimensional cocycle satisfying a lattice local central limit theorem, while the linear case uses Gaussian automorphisms with singular spectral measures.

What carries the argument

For the higher-degree theorem, the load-bearing object is a two-dimensional cocycle $S_n(f)$ built from i.i.d. blocks, satisfying a lattice local central limit theorem and, newly, a distinct-visits property: for every $N$, the increments $\{S_{j+N}(f)-S_N(f): -N\le j\le N\}$ take $2N+1$ distinct lattice points on a set of positive measure (Proposition 3.7). The proof obtains this by forcing a lexicographically increasing run in the first coordinate, using block independence on the long range $0\le j\le 2d_k+p_k$. Around this cocycle the authors form a skew product $\tilde T$ over a zero-entropy base, then define a permutation $\pi_y:\mathbb{Z}^2\to\mathbb{Z}^2$ that transports the $p_2$-return values $S_{p_2(n)}(f)(y)$ to the $p_1$-return values for every $n$ in a density-one set $K_y$; conjugating $\tilde T$ by $\pi_y$ yields $\tilde S$, and in a $k$-fold product the two dynamics flip the same coordinate bit in opposite directions. For the linear theorem, the machinery is instead a unitary reflection $W = P - (I-P)$ on the Gaussian space, where $P$ is projection onto the span of the Gaussian variable; the transformation $S$ has Koopman operator $V = WUW$, the intersection measure decays like $|\hat\sigma(n)|^{2k}$ with polynomial Fourier decay, and $k$ is chosen so that the probabilities are summable.

What would settle it

Compute the measure of the event $0<S_1(f)<\cdots<S_{2N}(f)$ in the lexicographic order for the explicit block cocycle from Proposition 3.6 with $N=1$; Proposition 3.7 declares this measure positive, and if it were zero the permutation construction in Lemma 3.4 would collapse.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the obstruction to double recurrence in the non-commuting zero-entropy world is degree parity: if both iterates are linear or both have degree at least two, the recurrence conclusion of the commuting polynomial theorem is false in the strongest sense. Theorem 1.1 asserts that for injective $p_1,p_2 \in \mathbb{Z}[t]$ with $p_1(0)=p_2(0)=0$ and $\deg p_i \ge 2$, there exist mixing zero-entropy transformations $T,S$ and a set $A$ with $\mu(A)>0$ such that $A \cap T^{-p_1(n)}A \cap S^{-p_2(n)}A = \varnothing$ for all $n \in \mathbb{N}$; Theorem 1.2 is the same statement for $p_i(n)=c_i n$ with nonzero integers $c_i$. The constructions are explicit: a skew product driven by a $\mathbb{Z}^2$-valued cocycle with an i.i.d.-block structure, conjugated by a coordinate permutation that swaps the two polynomial-return sets, and, for linear iterates, $k$-fold products of Gaussian automorphisms whose spectral measures have polynomial Fourier decay and which possess roots of all orders. The conclusion is a complete answer to the question posed in [FH23]: the recurrence theorem for the pair $(n,p(n))$ with $\deg p \ge 2$ marks exactly the boundary of the phenomenon.

Load-bearing premise

The load-bearing premise is that the block independence of the two-dimensional cocycle holds over the long range $0\le j\le 2d_k+p_k$ (and, for the linear theorem, that singular spectral measures with polynomial Fourier decay exist); if either fails, the positive-measure set witnessing empty intersections may not exist.

Editorial extensions

If this is right

For every injective integer polynomial pair $p_1,p_2$ with $p_1(0)=p_2(0)=0$ and $\deg p_1,\deg p_2\ge2$, there exist mixing, zero-entropy transformations $T,S$ and a positive-measure set $A$ with $A\cap T^{-p_1(n)}A\cap S^{-p_2(n)}A=\varnothing$ for every $n\in\mathbb{N}$.
For every pair of nonzero integers $c,d$, the same total failure holds for the linear iterates $cn$ and $dn$, extending the previously known $n,n$ case.
Together with the positive result in [FH23], the motivating question is now answered completely: the only pair type for which double recurrence is guaranteed in this setting is one linear iterate together with one higher-degree iterate.
The constructed counterexamples are mixing and have zero entropy, so the failure is not an artifact of poor mixing or positive entropy; it is intrinsic to non-commuting deterministic pairs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The long-range block-independence property behind Proposition 3.7 should transfer to other cocycle questions: whenever a zero-entropy system carries a cocycle whose blocks are independent over ranges longer than the block length, bounded windows of partial sums visit distinct lattice points with positive probability, which is exactly the input needed for further non-convergence examples.
Because the Gaussian construction gives roots of all orders, the same method likely handles linear iterates with rational slopes, or even pairs of affine sequences, by reparametrizing the embedded flow; the paper only states integer coefficients $c,d$.
A natural stress test is to push the same equal-degree construction to $d$-tuples of polynomials: the $k$-fold product trick suggests multiple recurrence fails for any tuple of injective equal-degree polynomials, and the mixed-degree boundary from [FH23] could be explored in higher-order averages.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline is that this is a genuine advance. Theorems 1.1 and 1.2 together with the FH23 positive result give the full classification for injective polynomial iterates vanishing at zero: recurrence holds exactly when one polynomial is linear and the other has degree at least two, and fails in the two-linear and two-high-degree cases. The construction for degree at least two is the substantive new work, and the cocycle-range positivity result in Section 3.2 is a real new tool, not a repackaging of [KS24]. The Gaussian treatment of the linear case in Section 4 is clean, self-contained, and handles arbitrary nonzero c,d via roots of Gaussian automorphisms.

I agree with the reader that the central arguments are coherent. I also checked the stress-test concern about Proposition 3.7 and do not think it lands. The needed ranges work out: Z(n) is a function of the previous levels on [0,2d_K+p_K], and for each tail level k≥K the shifts used by Y(n) lie inside [0,2d_k+p_k]. So Proposition 3.6(c), exactly as stated, gives the required independence by induction over tail levels. The proof is terse and would benefit from spelling that induction out, but the independence claim is supported by the quoted block-independence.

The real soft spots are two fixable gaps. First, Lemma 3.10 states (1-alpha^2)^L >= exp(-alpha^2 L), which has the wrong direction; the valid bound with an extra factor of 2 still gives a summable product, so nothing breaks. There is also a missing factor of 2 in the exponent for D_k, again harmless. Second, in Section 2.1 and again in Section 4, the passage from positive-measure D to a set A with literally empty intersections is too quick: the maximality of M only gives measure zero, not emptiness, when n+M is still small. The standard fix is to remove the countable union of the measure-zero intersections from A; this is a routine repair.

The reliance on [KS24] is substantial, especially for the 2D LCLT and the block-independence proposition. That is not circular, because the new positivity result is proved here, and the Gaussian section is independent. But referees should check that [KS24] actually proves Proposition 3.6(c) in the stated long-window form, since the paper explicitly depends on it.

Who is this for? Ergodic theorists and ergodic Ramsey theorists working on non-commuting multiple recurrence. It deserves a serious referee and, with the small repairs above, should be publishable in a good journal. I would send it to peer review rather than desk reject.

Referee Report

3 major / 6 minor

Summary. The paper constructs counterexamples to double recurrence for non-commuting, mixing, zero-entropy measure-preserving transformations. Theorem 1.1 handles the case where both iterates are injective integer polynomials of degree at least 2 vanishing at 0, and Theorem 1.2 handles the case where both iterates are linear (c n and d n). Together with an earlier positive result of Frantzikinakis and Host for one linear and one higher-degree iterate, the paper claims a complete answer to the double-recurrence question in the zero-entropy non-commuting setting. The higher-degree construction uses a 2-dimensional local central limit theorem cocycle from [KS24] and a new positivity result for the range of the cocycle; the linear construction uses Gaussian automorphisms built from singular measures with polynomial Fourier decay.

Significance. If the proofs are correct, the results settle the remaining cases of a natural question of Frantzikinakis and Host and are therefore significant for the field of non-commuting multiple recurrence. The Gaussian construction in Section 4 is elegant and largely self-contained, relying only on standard facts about Gaussian automorphisms and the known existence of singular measures with polynomial Fourier decay. The new positivity result Theorem 3.1(b) is of independent interest. The paper is clearly written and the main ideas are well motivated, with the simple n,n case presented first as a baby model.

major comments (3)

[§3.2, Lemma 3.10] The displayed inequality (1 - α_k^2)^{p_k + 2N} ≥ exp(-α_k^2(p_k + 2N)) is false for 0 < α_k < 1, since log(1-x) < -x for x > 0. The subsequent estimate exp(-α_k^2(p_k+2N)) ≥ exp(-2/p_k) therefore does not follow from the previous line. The lemma is repairable: using the standard bound (1-x)^m ≥ exp(-2mx) for x ≤ 1/2 yields exp(-4/p_k) instead, and the product over k of exp(-4/p_k) still converges because ∑ 1/p_k < ∞. Nevertheless, the proof as written contains a false inequality in a key estimate and must be corrected.
[§2.1 and §4, trimming argument] The proof that the set A satisfies A ∩ T^{-n}A ∩ S^{-n}A = ∅ for all n ∈ N is incomplete. From the maximality of M, one obtains that for m > M with m ≤ N, the set D ∩ T^{-m}D ∩ S^{-m}D has measure zero, not that it is empty. Therefore the containment A ∩ T^{-n}A ∩ S^{-n}A ⊂ D ∩ T^{-(n+M)}D ∩ S^{-(n+M)}D only shows that the intersection is contained in a null set for n+M ≤ N, not that it is empty. The argument can be fixed by removing the null set ⋃_{m=M+1}^{N} (D ∩ T^{-m}D ∩ S^{-m}D) from D before defining A, but this step is missing. The same gap appears in the Gaussian proof in Section 4 and affects the proof of Theorem 1.2 in both cases.
[§3.2, Proposition 3.6(c) and Proposition 3.7] The proof of the new positivity result Proposition 3.7 and of Lemma 3.10 relies critically on Proposition 3.6(c) as quoted from [KS24], where the previous levels {f̄_l, l<k} are evaluated on the long window 0 ≤ j ≤ 2d_k + p_k. This is a stronger independence statement than the 'own-window' independence that a natural block construction would give. If [KS24] only proves the weaker statement, then the independence of the sets {D_k} in Lemma 3.10 and the independence of Z(n) and Y(n) in Proposition 3.7 fail, and the proof of Theorem 1.1 collapses. The authors should either prove Proposition 3.6(c) in this paper or give a precise reference and verify that [KS24] indeed contains exactly this statement. This is the load-bearing point for Theorem 1.1.

minor comments (6)

[Introduction] The polynomial ring is denoted Z(t) in two places; this should be Z[t], since Z(t) conventionally denotes the field of rational functions.
[§2.1] The symbol B is used both for the product sigma-algebra B = D^{⊗3} and for the set B = (Y × [1]_0)^{⊗3}. This overloading is confusing in the proof of Theorem 1.2 (c=d=1); please use different letters.
[§3.2, Proposition 3.7] The notation K(n) is used in the sentence 'Z(n) is a function of {f̄_k^{(1)}∘T^j : 1 ≤ k < K(n), 0 ≤ j ≤ 2d_K + p_K}' even though K was defined earlier as K(N); also the claim that all shifts appearing in Z(n) lie in the stated range is not justified and should be spelled out.
[§3.2, Lemma 3.10] The symbol K(N) is reused with two different meanings: in Proposition 3.7 it is the smallest integer with 2N < p_K, while in Lemma 3.10 it is the smallest integer larger than κ with 2p_K < d_K. Please use different names to avoid ambiguity.
[§3, Section 3.2] The function f is defined as f = 2g in Section 3 and then redefined in Section 3.2 as the [KS24] LCLT function. Since scaling by 2 preserves the range-distinctness property needed for Theorem 3.1(b), the two uses are compatible, but this should be stated explicitly to avoid confusion.
[§3.2, Lemma 3.10] The sentence 'Note that as p_k + 2N < 2p_k there is no contradiction in the definition of D_k' is terse. What is actually needed is d_k > p_k + 2N, which follows from the defining condition 2p_K < d_K; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new positivity result is proved from the cited block-independence construction, not assumed, and the Gaussian argument in Section 4 is independent.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 is established by an explicit construction: the two-dimensional cocycle with block-independence is imported from the authors' prior work [KS24] (quoted as Proposition 3.6), but the new positivity result Theorem 3.1(b) / Proposition 3.7 is proved in Section 3.2 rather than assumed. It is derived from Proposition 3.6(b)-(c) together with Lemma 3.10, and Lemma 3.4 and Proposition 3.5 then use Theorem 3.1(b) to obtain the positive-measure set C. The permutation pi_y is an explicit construction, not a fitted or pre-assumed input. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked to forbid alternatives. The linear case Theorem 1.2, proved in Section 4, is independent of the block-independence input: it relies on external results on singular measures with polynomial Fourier decay [Kau80] and standard Gaussian automorphism facts about roots and flows. The paper does depend heavily on self-cited results from [KS24] and [KV22], but that dependence is a citation of prior theorems, not a reduction of the target statement to itself. The skeptical concern about the exact range in Proposition 3.6(c) is a correctness or robustness risk, not a circularity, because the paper does not assume the conclusion of Theorem 1.1 or Theorem 3.1(b). Therefore no specific circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs import the local limit theorems and block-independence results from [KV22] and [KS24], plus Gaussian machinery and Kaufman's singular measures. No ad hoc numerical parameters are fit; constants k, N, C, M are existential choices. The new content is the positivity of the cocycle range event and the extension to linear coefficients.

assumptions (5)

domain assumption Existence of 1D lattice LCLT function for any ergodic aperiodic T (KV22 Theorem 2.1)
Used in Section 2 to construct f with m(S_n(f)=j) approximately (2 pi sigma^2 n)^{-1/2} exp(-j^2/2n sigma^2); cited to [KV22], not proved here.
domain assumption Existence of 2D LCLT function with i.i.d. block structure (KS24 Theorem 2.4 and Proposition 3.6)
The core dependency for Theorem 1.1; Section 3.2 proves an additional positivity property but relies on the block independence from [KS24].
domain assumption Existence of singular measures on the circle with polynomial Fourier decay (Kaufman 1980)
Used in Theorem 4.1 and Theorem 1.2 to get Gaussian automorphisms with mixing, zero entropy and root powers; cited to [Kau80].
domain assumption Gaussian automorphisms have roots of all orders, zero entropy for singular spectral measures, and second quantization lifting
Used in Section 4 to pass from P, Q to L, R and to build S via unitary equivalence; cited to [CFS82], [Pin60], [dlR93], [JRDLR23].
standard math Abramov-Rokhlin formula for entropy of skew products
Used to conclude skew products in Proposition 2.2 and Section 3 have zero entropy; referenced to [HSY24b] and [KS24].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Counterexamples to double recurrence for non-commuting deterministic transformations." pith.science (2026). https://pith.science/paper/5B5XG3KC

@misc{pith2026250715528,
  author       = {Pith},
  title        = {Pith review of: Counterexamples to double recurrence for non-commuting deterministic transformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5B5XG3KC}},
  note         = {Machine review of arXiv:2507.15528}
}

abstract

We show that if $p_1,p_2$ are injective, integer polynomials that vanish at the origin, such that either both are of degree $1$ or both are of degree $2$ or higher, then double recurrence fails for non-commuting, mixing, zero entropy transformations. This answers a question of Frantzikinakis and Host completely.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 22 canonical work pages

[1]

Terrence M. Adams. Smorodinsky's conjecture on rank-one mixing. Proc. Amer. Math. Soc. , 126(3):739--744, 1998

work page 1998
[2]

Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures

Amir Algom , Federico Rodriguez Hertz , and Zhiren Wang . Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures . arXiv e-prints , page arXiv:2306.01275, June 2023

arXiv 2023
[3]

Non-convergence of some non-commuting double ergodic averages

Tim Austin. Non-convergence of some non-commuting double ergodic averages. Proc. Amer. Math. Soc. , 153(4):1701--1707, 2025

work page 2025
[4]

Polynomial F ourier decay for fractal measures and their pushforwards

Simon Baker and Amlan Banaji. Polynomial F ourier decay for fractal measures and their pushforwards. Math. Ann. , 392(1):209--261, 2025

work page 2025
[5]

Bergelson and A

V. Bergelson and A. Leibman. Polynomial extensions of van der W aerden's and S zemer\' e di's theorems. J. Amer. Math. Soc. , 9(3):725--753, 1996

work page 1996
[6]

Bergelson and A

V. Bergelson and A. Leibman. A nilpotent R oth theorem. Invent. Math. , 147(2):429--470, 2002

work page 2002
[7]

I. P. Cornfeld, S. V. Fomin, and Ya. G. Sina . Ergodic theory , volume 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinski

work page 1982
[8]

Darren Creutz and Cesar E. Silva. Mixing on rank-one transformations. Studia Math. , 199(1):43--72, 2010

work page 2010

Show all 23 references

[9]

Entropie d'un syst\`eme dynamique gaussien: cas d'une action de Z ^d

Thierry de la Rue. Entropie d'un syst\`eme dynamique gaussien: cas d'une action de Z ^d . C. R. Acad. Sci. Paris S\'er. I Math. , 317(2):191--194, 1993

1993
[10]

Multiple recurrence and convergence without commutativity

Nikos Frantzikinakis and Bernard Host. Multiple recurrence and convergence without commutativity. J. Lond. Math. Soc. (2) , 107(5):1635--1659, 2023

2023
[11]

Furstenberg and Y

H. Furstenberg and Y. Katznelson. An ergodic S zemer\' e di theorem for commuting transformations. J. Analyse Math. , 34:275--291 (1979), 1978

1979
[12]

Furstenberg

H. Furstenberg. Recurrence in ergodic theory and combinatorial number theory . Princeton University Press, Princeton, NJ, 1981. M. B. Porter Lectures

1981
[13]

A counterexample on multiple convergence without commutativity

Wen Huang , Song Shao , and Xiangdong Ye . A counterexample on multiple convergence without commutativity . arXiv e-prints , page arXiv:2407.10728, July 2024

2024 arXiv
[14]

A counterexample on polynomial multiple convergence without commutativity

Wen Huang, Song Shao, and Xiangdong Ye. A counterexample on polynomial multiple convergence without commutativity. Bull. Soc. Math. France , 152(1):149--168, 2024

2024
[15]

Dynamical systems of probabilistic origin: G aussian and P oisson systems

\'Elise Janvresse, Emmanuel Roy, and Thierry De La Rue. Dynamical systems of probabilistic origin: G aussian and P oisson systems. In Ergodic theory , Encycl. Complex. Syst. Sci., pages 217--232. Springer, New York, [2023] 2023

2023
[16]

Fourier transforms of G ibbs measures for the G auss map

Thomas Jordan and Tuomas Sahlsten. Fourier transforms of G ibbs measures for the G auss map. Math. Ann. , 364(3-4):983--1023, 2016

2016
[17]

R. Kaufman. Continued fractions and F ourier transforms. Mathematika , 27(2):262--267, 1980

1980
[18]

Alexander S. Kechris. Global aspects of ergodic group actions , volume 160 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2010

2010
[19]

Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages, 2024

Zemer Kosloff and Shrey Sanadhya. Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages, 2024

2024
[20]

Local limit theorem in deterministic systems

Zemer Kosloff and Dalibor Voln\'y. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincar\' e Probab. Stat. , 58(1):548--566, 2022

2022
[21]

M. S. Pinsker. Dynamical systems with completely positive or zero entropy. Sov. Math., Dokl. 1 , pages 937--938 (English), 1960

1960
[22]

Ryzhikov

Valery V. Ryzhikov . Non-commuting transformations with non-converging 2-fold ergodic averages . arXiv e-prints , page arXiv:2407.13741, July 2024

2024 arXiv
[23]

Ryzhikov

Valery V. Ryzhikov. Spectra and joint dynamics of poisson suspensions for rank-one automorphisms, 2024

2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Terrence M. Adams. Smorodinsky's conjecture on rank-one mixing. Proc. Amer. Math. Soc. , 126(3):739--744, 1998

work page 1998

[2] [2]

Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures

Amir Algom , Federico Rodriguez Hertz , and Zhiren Wang . Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures . arXiv e-prints , page arXiv:2306.01275, June 2023

arXiv 2023

[3] [3]

Non-convergence of some non-commuting double ergodic averages

Tim Austin. Non-convergence of some non-commuting double ergodic averages. Proc. Amer. Math. Soc. , 153(4):1701--1707, 2025

work page 2025

[4] [4]

Polynomial F ourier decay for fractal measures and their pushforwards

Simon Baker and Amlan Banaji. Polynomial F ourier decay for fractal measures and their pushforwards. Math. Ann. , 392(1):209--261, 2025

work page 2025

[5] [5]

Bergelson and A

V. Bergelson and A. Leibman. Polynomial extensions of van der W aerden's and S zemer\' e di's theorems. J. Amer. Math. Soc. , 9(3):725--753, 1996

work page 1996

[6] [6]

Bergelson and A

V. Bergelson and A. Leibman. A nilpotent R oth theorem. Invent. Math. , 147(2):429--470, 2002

work page 2002

[7] [7]

I. P. Cornfeld, S. V. Fomin, and Ya. G. Sina . Ergodic theory , volume 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinski

work page 1982

[8] [8]

Darren Creutz and Cesar E. Silva. Mixing on rank-one transformations. Studia Math. , 199(1):43--72, 2010

work page 2010