Veech's question: density of pure point induced transformations of an irrational rotation

For a fixed irrational rotation $T$ of the circle, consider all induced (first-return) maps $T_A$ on measurable sets $A$ of positive measure, topologized through the metric $d(A,B)=m(A\triangle B)$ on the measure algebra. A question of William Veech, recorded by Petersen (1973) and Conze (1972) and recalled by Knill (2023), asks whether the class of sets $A$ for which $T_A$ has pure point spectrum is dense in this metric. Since the induced maps of an ergodic transformation are generically weakly mixing, this pure point class is meager; the question is whether it is nevertheless dense, as the rationals are in the reals. A positive answer, combined with Simon's Wonderland theorem, would force the generic induced system of a rotation to have purely singular continuous spectrum. The question extends verbatim to any ergodic transformation with pure point spectrum. No resolution of the density question is known in the literature to date.

Background

Let $T$ be a measure-preserving invertible transformation (automorphism) of a Lebesgue probability space $(X,\mathcal{A},m)$, that is, a standard non-atomic probability space such as the circle with Lebesgue measure. The map $T$ is ergodic if every $T$-invariant measurable set has measure $0$ or $1$. The guiding example is the irrational rotation $T(x)=x+\alpha \bmod 1$ of the circle $\mathbb{T}$, which is ergodic exactly when $\alpha$ is irrational.

For a measurable set $A$ with $m(A)>0$, the induced or first-return map $T_A$ is the restriction of the dynamics to $A$ accelerated to successive returns: $T_A(x)=T^{r_A(x)}(x)$, where $r_A(x)=\min\{k\ge 1 : T^k x\in A\}$ is the return-time function. This construction, going back to the 1940s, is the measurable analogue of a Poincaré return map; the original system $T$ can be rebuilt as a Kakutani skyscraper — a tower whose levels are the successive translates of $A$ between returns — over $T_A$ with roof function $r_A$. Modding out null sets, the family of measurable sets carries the complete separable metric $d(A,B)=m(A\triangle B)$ (symmetric difference), so questions about the family of induced maps $T_A$ become Baire-category questions on this Polish space, the measure algebra.

The spectral side: to any measure-preserving $T$ one associates the Koopman unitary operator $U_T:f\mapsto f\circ T$ on $L^2(X,m)$. The transformation $T$ has pure point (discrete) spectrum if $L^2$ admits an orthonormal basis of $U_T$-eigenfunctions; by the classical characterization, an ergodic transformation has pure point spectrum exactly when it is measurably isomorphic to an ergodic rotation on a compact abelian group, so for ergodic systems pure point spectrum and being a rotation are synonymous. At the opposite pole, $T$ is weakly mixing when it has no nonconstant eigenfunctions. Entropy scales under inducing by the standard law $h(T_A)=h(T)/m(A)$, so every induced map of an irrational rotation has zero entropy.

What is known about induced maps of a fixed ergodic $T$: weakly mixing induced maps are dense (a classical density result recounted by Knill), and moreover the set $\{A : T_A \text{ weakly mixing}\}$ is residual, that is, comeager — a statement that del Junco and Rudolph attribute to Conze (1972) and that Knill also credits to a 1973 remark of Petersen. Because a weakly mixing map has no nontrivial point spectrum, the class $\mathcal{P}=\{A : T_A \text{ has pure point spectrum}\}$ is meager; it is nonempty, since $A=X$ gives $T_A=T$, and inducing a rotation on suitable intervals produces exchanges of two intervals, which are themselves rotations. del Junco and Rudolph (1996) sharpened the residual picture: for $T$ of zero entropy that is loosely Bernoulli — the zero-entropy analogue of the Bernoulli property, defined by an orbit-name matching condition that irrational rotations satisfy — a residual set of $A$ yields $T_A$ rank-one (generable from a single cutting-and-stacking tower) and rigid (some powers of its Koopman operator converge strongly to the identity); for positive-entropy $T$ the generic induced map is Bernoulli or K. Hansel (1973) showed a complementary abundance: for any aperiodic $T$, any prescribed $\lambda$ on the unit circle, and any $\varepsilon>0$, there is a set $A$ with $m(A)\ge 1-\varepsilon$ such that $\lambda$ is an eigenvalue of $T_A$ — but no way is known to upgrade individual eigenvalues to a complete eigenbasis.

The question below asks whether the meager class $\mathcal{P}$ is nonetheless dense. The stakes are set by Simon's Wonderland theorem (1995): in a Polish family of unitary operators in which pure point spectrum is dense and no member has absolutely continuous spectrum, the generic member has purely singular continuous spectrum. Knill observes that a positive answer for a rotation $T$ would transfer this conclusion to the family of induced Koopman operators, forcing purely singular continuous spectrum for the generic induced system — an expected but unproved behavior.

Problem Statement

Let $T(x)=x+\alpha \bmod 1$ be an irrational rotation of the circle $\mathbb{T}$ (so $\alpha$ is irrational), acting on the Lebesgue probability space $(\mathbb{T},\mathcal{A},m)$, and for a measurable set $A$ with $m(A)>0$ let $T_A$ denote the induced (first-return) map on $(A,m/m(A))$, defined by $T_A(x)=T^{r_A(x)}(x)$ with return time $r_A(x)=\min\{k\ge 1 : T^k x\in A\}$. Veech's question, as recorded by Petersen (1973, p. 229) and Conze (1972) and recalled by Knill (2023):

Is the class $\mathcal{P}=\{A\in\mathcal{A} : m(A)>0 \text{ and } T_A \text{ has pure point spectrum}\}$ dense in the measure algebra $(\mathcal{A},d)$ — measurable sets modulo null sets — with $d(A,B)=m(A\triangle B)$?

Equivalently: for every measurable set $B$ (of arbitrary measure, including measure zero) and every $\varepsilon>0$, must there exist a measurable set $A$ with $m(A)>0$ and $m(A\triangle B)<\varepsilon$ such that the Koopman operator of $T_A$ (the unitary operator $f\mapsto f\circ T_A$ on $L^2(A,m/m(A))$) admits a complete orthonormal basis of eigenfunctions — equivalently, since $T_A$ is ergodic whenever $T$ is, such that $T_A$ is measurably isomorphic to an ergodic rotation on a compact abelian group? Knill records the same question for an arbitrary ergodic transformation with pure point spectrum, and notes that a positive answer, combined with Simon's Wonderland theorem, would force the generic induced system $T_A$ to have purely singular continuous spectrum. The problem is to prove or refute the density claim for each irrational rotation; as Knill notes, the same question can also be asked in the extended setting of an arbitrary ergodic transformation with pure point spectrum.

References

1. Oliver Knill, "On the cohomology of measurable sets", arXiv:2307.12476 (2023), DOI: 10.48550/arXiv.2307.12476, https://arxiv.org/abs/2307.12476
2. Karl Petersen, "Spectra of induced transformations", in Recent Advances in Topological Dynamics (Proc. Conf. Topological Dynamics, Yale Univ. 1972), Lecture Notes in Mathematics 318, pp. 226-230, Springer, 1973, DOI: 10.1007/bfb0061740, https://doi.org/10.1007/bfb0061740
3. Jean-Pierre Conze, "Équations fonctionnelles et systèmes induits en théorie ergodique", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 23 (1972), 75-82, DOI: 10.1007/BF00536692, https://doi.org/10.1007/BF00536692
4. G. Hansel, "Automorphismes induits et valeurs propres", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 25 (1973), 155-157, DOI: 10.1007/BF00533101, https://doi.org/10.1007/BF00533101
5. Andres del Junco and Daniel J. Rudolph, "Residual behavior of induced maps", Israel Journal of Mathematics 93 (1996), 387-398, DOI: 10.1007/BF02761114, https://doi.org/10.1007/BF02761114
6. Barry Simon, "Operators with singular continuous spectrum: I. General operators", Annals of Mathematics (2) 141, No. 1 (1995), 131-145, DOI: 10.2307/2118629, https://doi.org/10.2307/2118629
7. Alessandro Carderi, Matthieu Joseph, François Le Maître, and Romain Tessera, "Belinskaya's theorem is optimal", Fundamenta Mathematicae 263, No. 1 (2023), 51-90, DOI: 10.4064/fm266-4-2023, arXiv:2201.06662, https://arxiv.org/abs/2201.06662
8. Daniel J. Rudolph and Bethany D. Springer, "Nearly Continuous Even Kakutani Equivalence of Strongly Rank One Transformations", arXiv:1211.2297 (2012), https://arxiv.org/abs/1211.2297