Strongly expansive minimal models for $\times p_1,\dots,\times p_d$-ergodic measures and the higher-dimensional complexity gap (Cyr–Kra Conjecture 3.2)

For multiplicatively independent integers $p_1,\dots,p_d$ with $d>2$ and a nonatomic Borel probability measure $\mu$ on $[0,1)$ ergodic under the joint action of the maps $x\mapsto p_i x \pmod 1$, Cyr and Kra asked whether the natural extension admits a symbolic topological model whose pattern complexity is forced to grow at least on the order of $n^d$ — the higher-dimensional analogue of their two-dimensional complexity gap, which showed that every Jewett–Krieger model of a nonatomic $\times p,\times q$-ergodic measure satisfies $\liminf_n P(R_n)/n^2 \ge 1/2$. They reduced the generalization to Conjecture 3.2: every such $\mu$ admits a minimal topological model that is strongly expansive (a coordinate axis of the model determines the entire configuration), since a theorem of the same authors then yields $\liminf_n P(n)/n^d > 0$. Neither Conjecture 3.2 nor the higher-dimensional complexity gap has been resolved or refuted; the problem remains open.

Background

For an integer $p\ge 2$, let $\times p$ denote the expanding circle map $x \mapsto px \pmod 1$ on $[0,1)$. Integers $p_1,\dots,p_d \ge 2$ are multiplicatively independent when they admit no nontrivial multiplicative relations, i.e. $p_1^{a_1}\cdots p_d^{a_d}=1$ with integers $a_i$ forces all $a_i=0$ (for two integers this says they are not both powers of a common integer). A landmark theorem of Furstenberg (1967) shows that a closed subset of $[0,1)$ invariant under two multiplicatively independent maps $\times p$ and $\times q$ is either finite or all of $[0,1)$, and he conjectured the analogous measure-theoretic dichotomy: a Borel probability measure invariant under both $\times p$ and $\times q$ and ergodic for the joint $\mathbb{Z}^2$-action they generate is either Lebesgue measure or atomic. The positive-entropy case was settled by Rudolph for coprime $p,q$ and by Johnson for multiplicatively independent $p,q$: if the measure-theoretic entropy of one of the maps is positive, the measure is Lebesgue. The zero-entropy case — whether nonatomic zero-entropy jointly ergodic measures exist at all — remains one of the most famous open problems in ergodic theory; recent work still treats it as open, e.g. a 2023–2024 characterization of the conjecture as a uniqueness statement for traces on a group $C^*$-algebra.

One way to attack the problem is symbolic. Passing to the natural extension (the standard minimal invertible cover of a non-invertible system, here carrying a free $\mathbb{Z}^2$-action generated by commuting lifts of $\times p$ and $\times q$), the Jewett–Krieger theorem — extended to $\mathbb{Z}^d$-actions by Weiss and refined by Rosenthal — guarantees a topological model: a minimal, uniquely ergodic $\mathbb{Z}^d$-subshift (a closed, shift-invariant subset of $\mathcal{A}^{\mathbb{Z}^d}$ over a finite alphabet) that is measure-theoretically isomorphic to the given measure-preserving system. For a subshift $X \subset \mathcal{A}^{\mathbb{Z}^d}$, the pattern complexity $P_X(n,\dots,n)$ counts the distinct $n\times\cdots\times n$ blocks occurring in configurations of $X$; it is a basic combinatorial invariant (bounded iff $X$ is finite, exponential iff the topological entropy is positive). The classical Morse–Hedlund theorem says a one-dimensional sequence with complexity $P(n)\le n$ is periodic, and Nivat conjectured in 1997 that a two-dimensional configuration with $P(n,k)\le nk$ for some $n,k$ is periodic; Cyr and Kra proved the partial result that a transitive $\mathbb{Z}^2$-subshift with $P(n,k)\le nk/2$ has a nontrivial period vector.

Combining these ingredients, Cyr and Kra proved their two-dimensional complexity gap: if $\mu$ is nonatomic and ergodic for the joint $\times p,\times q$-action, then every Jewett–Krieger model $\widehat X$ of its natural extension satisfies $\liminf_n P_{\widehat X}(R_n)/n^2 \ge 1/2$; equivalently, although aperiodic strictly ergodic (i.e. minimal and uniquely ergodic) $\mathbb{Z}^2$-subshifts of subquadratic complexity do exist, none of them can model a nonatomic jointly ergodic $\times p,\times q$ measure. This rules out an entire family of combinatorially simple potential counterexamples to Furstenberg's conjecture.

In the closing section of that paper, the authors pose the higher-dimensional generalization: for $d>2$ and $p_1,\dots,p_d$ multiplicatively independent, can the natural extension of a nonatomic $\times p_1,\dots,\times p_d$-ergodic measure have a topological model whose complexity is $o(n^d)$? The two-dimensional argument is blocked in higher dimensions by an unpublished 2006 construction of J. Cassaigne (a counterexample to a conjecture of Lagarias and Pleasants on repetitive low-complexity sets): for $d>2$ there exist minimal $\mathbb{Z}^d$-subshifts, aperiodic in every direction, with $P_X(n,\dots,n) = o(n^{2+\varepsilon})$ for every $\varepsilon>0$, so low complexity alone no longer forces periodicity. However, the same authors proved (in their study of automorphism groups of minimal shifts of stretched exponential growth) that a one-dimensional minimal subshift carrying commuting homeomorphisms generating a $\mathbb{Z}^d$-action must satisfy $\liminf_n P(n)/n^d > 0$, and they note the minimality hypothesis can be relaxed to minimality of the $\mathbb{Z}^d$-system. This theorem applies exactly to models that are strongly expansive: a $\mathbb{Z}^d$-subshift in which some coordinate axis is strongly expansive, meaning that two configurations of the model that agree on that axis are equal, so that the restriction to the axis, together with commuting homeomorphisms reconstructing the remaining coordinates, determines the entire configuration. They therefore state Conjecture 3.2: every nonatomic $\times p_1,\dots,\times p_d$-ergodic measure admits a strongly expansive, minimal topological model; and they call this 'the only obstruction' to extending their complexity gap to dimensions $d>2$, since the conjecture combined with their theorem implies that every such system is measurably isomorphic to a subshift whose complexity grows on the order of $n^d$.

Problem Statement

Let $d>2$, let $p_1,\dots,p_d \ge 2$ be multiplicatively independent integers, and let $\mu$ be a nonatomic Borel probability measure on $[0,1)$ that is invariant under each map $x \mapsto p_i x \pmod 1$ and ergodic for the joint $\mathbb{Z}^d$-action they generate. Consider the natural extension of the system $([0,1),\mathcal B,\mu,\times p_1,\dots,\times p_d)$. Resolve the question posed by Cyr and Kra in Section 3 of 'Free ergodic $\mathbb{Z}^2$-systems and complexity', whose core is their Conjecture 3.2:

(A) Does there exist a strongly expansive, minimal topological model for the natural extension — that is, a minimal, uniquely ergodic $\mathbb{Z}^d$-system that is measure-theoretically isomorphic to the natural extension and that (after coding on a finite alphabet) has a coordinate axis with the property that two configurations of the model agreeing on that axis are equal, so that the restriction to the axis together with the commuting coordinate homeomorphisms determines the whole configuration?

(B) More broadly, establish or refute the higher-dimensional complexity gap that (A) is designed to deliver: by the authors' theorem, a positive answer to (A) implies that every such system is measurably isomorphic to a subshift whose complexity function grows on the order of $n^d$ (in particular $\liminf_n P(n,\dots,n)/n^d > 0$ for such a model); the direct two-dimensional analogue of the gap would additionally show that no topological model of the natural extension has complexity $o(n^d)$.

The two parts form one scientific objective — extending the two-dimensional complexity gap for jointly ergodic $\times p$-invariant measures to rank $d>2$ — with (A) being the source's identified sufficient route. The objective concerns every multiplicatively independent $p_1,\dots,p_d$ and every such nonatomic jointly ergodic $\mu$.

References

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