Everywhere failure of simultaneous recurrence for minimal systems with commutativity not assumed (Conjectures 1–3 of Huang–Shao–Ye, 2024)

For two homeomorphisms $T,S$ of the same compact metric space $X$ with $(X,T)$ and $(X,S)$ both minimal, Huang, Shao and Ye (2024) proved that a residual set of points $x$ returns to itself simultaneously under $S^n$, $T^{p_1(n)},\ldots,T^{p_d(n)}$ along a common subsequence, for any nonlinear integer-valued polynomials $p_i$ vanishing at $0$. The same paper poses three conjectures asserting that this residual-set phenomenon is essentially sharp in the minimal category: minimal pairs with commutativity not assumed can be built in which simultaneous recurrence fails at every single point — in the linear case ($T^n$ versus $S^n$), for arbitrary polynomial pairs $T^{p(n)}$ versus $S^{q(n)}$ with $\deg p,\deg q\ge 2$ and $p(0)=q(0)=0$, and in a mixed linear/quadratic triple pattern ($T^{n}$, $T^{2n}$, $S^{n^2}$). The problem is to resolve these three conjectures, each by an explicit construction of the required minimal pairs or by a proof that such pairs cannot exist.

Background

A topological dynamical system (t.d.s.) is a pair $(X,T)$ where $X$ is a compact metric space and $T:X\to X$ is a homeomorphism. The system is called minimal if every orbit $\{T^n x:n\in\mathbb{Z}\}$ is dense in $X$; a point $x$ is recurrent if $T^{n_i}x\to x$ along some sequence $n_i\to\infty$. Classical topological multiple recurrence (Furstenberg and Furstenberg–Weiss) concerns one map together with its iterates: for a minimal system, points $x$ exist for which the diagonal tuple $(x,\ldots,x)$ returns to itself under the product map $T\times T^2\times\cdots\times T^d$; Furstenberg asked in 1981 whether every system contains a point whose diagonal orbit under all such finite products is minimal (i.e. the orbit closure contains no proper closed invariant subset), and Huang–Shao–Ye showed in 2022 that a minimal weakly mixing system (one whose self-product $X\times X$ is transitive, i.e. has a dense orbit) can fail this at every point, answering Furstenberg's question negatively. A much-studied complementary line (Frantzikinakis–Host, 2023, and subsequent counterexample papers) treats measure-theoretic recurrence and convergence of multiple ergodic averages when the transformations do not commute. The present problem sits at the topological, noncommutative end of this line: here $T$ and $S$ are two homeomorphisms of the same space $X$, not assumed to commute, and both $(X,T)$ and $(X,S)$ are minimal. A point $x$ is simultaneously recurrent (for the linear pattern) if there is a sequence $n_i\to\infty$ with $T^{n_i}x\to x$ and $S^{n_i}x\to x$ at the same times, i.e. $(x,x)$ is a recurrent point of the product action $T\times S$. Without minimality it is classical (Furstenberg's 1981 book, p. 40) that such simultaneous recurrence can fail at every point; the question is whether this can happen inside the minimal category. Huang, Shao and Ye (arXiv:2409.07979, 2024) proved the positive residual result: for any two minimal systems on the same space there is a residual set (a countable intersection of open dense sets, i.e. a dense $G_\delta$) of points $x$ such that for every finite family of nonlinear integer-valued polynomials $p_1,\ldots,p_d$ with $p_j(0)=0$ one can find $n_i\to\infty$ with $S^{n_i}x\to x$ and $T^{p_j(n_i)}x\to x$ for all $j$. Here an integer-valued polynomial is a map $p:\mathbb{Z}\to\mathbb{Z}$ given by a polynomial expression (e.g. one with integer coefficients). A residual set can be topologically large while missing many, even all but a meager set of, points; the paper's Section 1.5 conjectures that minimal examples exist in which the simultaneous return phenomenon fails at every point, for several natural return-time patterns. Three instances are posed. Conjecture 1 is the linear case: minimal $(X,T)$ and $(X,S)$ such that for no $x\in X$ is $(x,x)$ recurrent under $T\times S$. Conjecture 2 is the polynomial case: for any pair of integer-valued polynomials $p,q$ vanishing at $0$ with $\deg p,\deg q\ge 2$, minimal $(X,T)$ and $(X,S)$ such that no $x\in X$ admits $n_i\to\infty$ with $T^{p(n_i)}x\to x$ and $S^{q(n_i)}x\to x$; note that degree $\ge 2$ excludes the linear polynomials of Conjecture 1, so the two conjectures are logically independent instances of one objective. Conjecture 3 is a mixed linear/quadratic triple pattern: minimal $(X,T)$ and $(X,S)$ such that no $x\in X$ admits $n_i\to\infty$ with $T^{n_i}x\to x$, $T^{2n_i}x\to x$ and $S^{n_i^2}x\to x$ simultaneously. The paper's own methods — induced orbit spaces, the polynomial saturation theorem, and Furstenberg disjointness (an independence condition under which, for minimal systems, the product system is again minimal) — are exactly what produce the residual set, and the authors isolate as the open frontier whether minimality forces simultaneous recurrence at some point or whether the residual conclusion is the best possible.

Problem Statement

Resolve the three conjectures posed in Section 1.5 of Huang–Shao–Ye, 'Multiple recurrence without commutativity' (arXiv:2409.07979). In all three, $T$ and $S$ denote two homeomorphisms of the same compact metric space $X$, the systems $(X,T)$ and $(X,S)$ are both minimal, and $T$ and $S$ are not assumed to commute. Conjecture 1 (linear pattern): there exist minimal systems $(X,T)$ and $(X,S)$ such that for every $x\in X$ the pair $(x,x)$ is not recurrent under the product action $T\times S$; equivalently, there is no sequence $n_i\to\infty$ with $T^{n_i}x\to x$ and $S^{n_i}x\to x$ simultaneously. Conjecture 2 (polynomial pattern): for every pair of integer-valued polynomials $p,q$ (polynomials taking integer values at every integer) with $p(0)=q(0)=0$ and $\deg p\ge 2$, $\deg q\ge 2$, there exist minimal systems $(X,T)$ and $(X,S)$ such that for every $x\in X$ there is no subsequence $n_i\to\infty$ with $T^{p(n_i)}x\to x$ and $S^{q(n_i)}x\to x$. Conjecture 3 (mixed linear/quadratic triple pattern): there exist minimal systems $(X,T)$ and $(X,S)$ such that for every $x\in X$ there is no subsequence $n_i\to\infty$ with $T^{n_i}x\to x$, $T^{2n_i}x\to x$ and $S^{n_i^2}x\to x$ simultaneously. The common objective is to determine whether the residual-set simultaneous recurrence theorem of the source paper is sharp in the minimal category: either construct, for each conjecture, the required pair of minimal systems and prove the everywhere-failure property, or prove that in some case every pair of minimal systems on the same space must contain a point with the corresponding simultaneous recurrence. Each of the three conjectures must receive a definite answer (affirmative construction or refutation); they are logically independent instances — Conjecture 2's degree restriction excludes the linear pattern of Conjecture 1, and Conjecture 3 mixes two iterates of $T$ with one of $S$ — so no one of them settles the others.

References

1. Wen Huang, Song Shao, Xiangdong Ye, 'Multiple recurrence without commutativity', preprint arXiv:2409.07979 (2024). DOI: 10.48550/arXiv.2409.07979, https://arxiv.org/abs/2409.07979 (source of Conjectures 1–3)
2. Wen Huang, Song Shao, Xiangdong Ye, 'Multiply minimal points for the product of iterates', Israel Journal of Mathematics 251 (2022), 541–565. DOI: 10.1007/s11856-022-2429-9, https://doi.org/10.1007/s11856-022-2429-9 (also arXiv:2103.16759, https://arxiv.org/abs/2103.16759)
3. Zijie Lin, Kangbo Ouyang, 'Minimal PI-systems with all points multiply minimal', Nonlinearity 39 (2026), no. 3, 035010. DOI: 10.1088/1361-6544/ae49d4, https://doi.org/10.1088/1361-6544/ae49d4 (preprint arXiv:2511.15335, titled 'Minimal PI-systems with all points are multiply minimal', https://arxiv.org/abs/2511.15335)
4. Nikos Frantzikinakis, Bernard Host, 'Multiple recurrence and convergence without commutativity', Journal of the London Mathematical Society (2) 107 (2023), 1635–1659. DOI: 10.1112/jlms.12721, https://doi.org/10.1112/jlms.12721
5. Harry Furstenberg, 'Poincaré recurrence and number theory', Bulletin of the American Mathematical Society 5 (1981), 211–234. DOI: 10.1090/s0273-0979-1981-14932-6, https://doi.org/10.1090/s0273-0979-1981-14932-6
6. Harry Furstenberg, 'Recurrence in Ergodic Theory and Combinatorial Number Theory', Princeton University Press, Princeton, NJ, 1981. DOI: 10.1515/9781400855162, https://doi.org/10.1515/9781400855162
7. Wen Huang, Song Shao, Xiangdong Ye, 'A counterexample on multiple convergence without commutativity', Proceedings of the American Mathematical Society 153 (2025), no. 12 (published online October 2025). DOI: 10.1090/proc/17387, https://doi.org/10.1090/proc/17387 (preprint arXiv:2407.10728, https://arxiv.org/abs/2407.10728)
8. Wen Huang, Song Shao, Xiangdong Ye, 'A counterexample on polynomial multiple convergence without commutativity', Bulletin de la Société Mathématique de France 152 (2024), 149–168. DOI: 10.24033/bsmf.2886, https://doi.org/10.24033/bsmf.2886
9. Daniel Glasscock, Andreas Koutsogiannis, Anh N. Le, Joel Moreira, Florian K. Richter, Donald Robertson, 'A structure theorem for polynomial return-time sets in minimal systems', preprint arXiv:2511.02080 (2025, revised May 2026), https://arxiv.org/abs/2511.02080