Baire-generic behavior of dilation orbits on spaces of invariant measures: residuality of non-equidistribution and of averaged equidistribution for arbitrary sequences and matrix pairs (Badea-Grivaux, Questions 5.5, 5.6, 5.8, 5.9)

For an integer $n\ge 2$ let $T_n$ denote the times-$n$ map $x\mapsto nx\bmod 1$ on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Badea and Grivaux (Discrete Analysis 2024) proved by Baire category arguments that for distinct integers $p,q\ge 2$ a generic (residual) continuous $T_p$-invariant probability measure $\mu$ on $\mathbb{T}$ has some large Fourier coefficients along $(q^n)$, so $(T_{q^n}\mu)$ does not converge weak-star to Lebesgue measure, disproving a conjecture of Furstenberg; they gave a sufficient congruence criterion (condition (H)) covering $(q^n)$ and linearly recurrent sequences, and a multidimensional version under hypotheses on the matrices (diagonalizability with no unit-modulus eigenvalue, plus coprimality of determinants or triangularity). This problem records the open core they explicitly left: (i) is non-equidistribution of $(T_{c_n}\mu)$ residual for every strictly increasing integer sequence $(c_n)$ (Question 5.5); (ii) determine, for an arbitrary strictly increasing $(c_n)$, whether averaged equidistribution of the Cesaro means $\frac{1}{N}\sum_{n<N}T_{c_n}\mu$ is residual — recorded by the source as widely open even for $(c_n)=(q^n)$ with $p,q$ multiplicatively independent (Question 5.6); and (iii-iv) the same two questions in dimension $d\ge 2$ for arbitrary integer matrix pairs $A,B$ with nonzero determinants, i.e. are the hypotheses of the multidimensional theorem necessary (Questions 5.8 and 5.9).

Background

For an integer $n\ge 2$, the times-$n$ map (also called the dilation or $\times n$ map) is $T_n:\mathbb{T}\to\mathbb{T}$, $T_n(x)=nx\bmod 1$, on the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$; it shifts the $n$-ary digit expansion of $x$. Two integers $p,q\ge 2$ are multiplicatively independent if $\log p/\log q\notin\mathbb{Q}$, i.e. no powers of $p$ and $q$ coincide. Write $\mathcal{P}(\mathbb{T})$ for the Borel probability measures on $\mathbb{T}$ with the weak-star ($w^*$) topology, in which $\mu_k\to\mu$ iff all Fourier coefficients $\hat\nu(a)=\int_{\mathbb{T}}z^a\,d\nu(z)$, $a\in\mathbb{Z}$, converge; $\mathcal{P}_p(\mathbb{T})$ is the closed subspace of $T_p$-invariant measures ($\mu=T_p\mu$), a compact metrizable space on which Baire category arguments apply, and $\mathcal{P}_{p,c}(\mathbb{T})$ its subset of continuous (non-atomic) measures. A subset of a Polish space is residual if it contains a countable intersection of dense open sets; residual sets are the topologically generic ones.

Furstenberg's 1967 rigidity program predicts that expansions in multiplicatively independent bases share no structure: the only infinite closed subset of $\mathbb{T}$ invariant under both $T_p$ and $T_q$ is $\mathbb{T}$ itself (a theorem), while the famed $\times p,\times q$ conjecture asserts that the only continuous probability measure invariant under both $T_p$ and $T_q$ is Lebesgue measure (Leb) — still open in general, with the major partial result of Rudolph (1990) covering ergodic measures of positive entropy. A natural strengthening, formulated as conjecture (C3) by Lyons (1988) and attributed to Furstenberg, predicted that already invariance under one map forces equidistribution under the other: for every $\mu\in\mathcal{P}_{p,c}(\mathbb{T})$, the push-forwards $T_{q^n}\mu$ converge $w^*$ to Leb; equivalently $\hat\mu(aq^n)\to 0$ for every $a\ne 0$. This statement implies the $\times p,\times q$ conjecture. Partial positive results covered special classes: Lyons and Feldman-Smorodinsky for non-degenerate $p$-Bernoulli measures (infinite convolutions of digit distributions), Host (1995) for ergodic measures of positive entropy when $\gcd(p,q)=1$, extended by Lindenstrauss (2001) to $p$ not dividing any power of $q$ and by Hochman-Shmerkin (2015) to all multiplicatively independent pairs; from these, Johnson and Rudolph (1995) deduced that for ergodic positive-entropy $\mu$, $T_{q^n}\mu\to\mathrm{Leb}$ along a subsequence of upper density one, and that the set of $\mu\in\mathcal{P}_{p,c}(\mathbb{T})$ equidistributing along such a large subsequence is residual.

Badea and Grivaux (2024) disproved conjecture (C3) in the strongest generic form: for any distinct $p,q\ge 2$ the set $S_{p,q}=\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;:\;\limsup_{n\to\infty}|\hat\mu(q^n)|>0\}$ is residual in $(\mathcal{P}_p(\mathbb{T}),w^*)$, so generically $T_{q^n}\mu\not\to\mathrm{Leb}$ (positive entropy plays no role in their proof). Their engine is a general criterion: if the strictly increasing integer sequence $(c_n)$ satisfies a congruence condition (H) — there are integers $h_l\ne 0$, $t_i$ and infinitely many $N$ with $h_lc_n\equiv t_i\pmod{p^N-1}$ for suitable $n$ — then

$$G_{p,(c_n)}:=\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;:\;T_{c_n}\mu\not\xrightarrow{\;w^*\;}\mathrm{Leb}\}$$

is residual. This covers $(q^n)$, linearly recurrent sequences $c_n=a_1c_{n-1}+\dots+a_Lc_{n-L}$ with $\gcd(a_L,p)=1$, and the classes treated earlier by Meiri and by Lindenstrauss-Meiri-Peres via their $p$-adic collision exponent. On the $d$-torus $\mathbb{T}^d$, for integer matrices $A,B$ with $\det A,\det B\ne 0$ and the endomorphism $T_A(x)=Ax\bmod 1$, where $\mathrm{Leb}_d$ is Lebesgue measure on $\mathbb{T}^d$ and $\mathcal{P}_{A,c}(\mathbb{T}^d)$ denotes the continuous (non-atomic) $T_A$-invariant measures, they proved the analogous residuality of

$$G_{A,B}:=\{\mu\in\mathcal{P}_{A,c}(\mathbb{T}^d)\;:\;T_{B^n}\mu\not\xrightarrow{\;w^*\;}\mathrm{Leb}_d\}$$

under the hypotheses that $A$ is diagonalizable over $\mathbb{C}$ with no eigenvalue of modulus $1$, and that either $\gcd(\det A,\det B)=1$ or $A$ is triangular. Together with Johnson-Rudolph, the emerging one-dimensional picture is subtle: a generic continuous $T_p$-invariant measure fails to equidistribute along the full sequence $(q^n)$ yet equidistributes along a subsequence of upper density one. The questions below ask whether this picture is universal. Note that residuality of the set of non-convergence and residuality of the set of Cesaro-averaged convergence are compatible, since convergence of the averages $\frac{1}{N}\sum_{n=0}^{N-1}T_{c_n}\mu$ does not imply convergence of $(T_{c_n}\mu)$ itself.

Problem Statement

Determine, with rigorous proof, the Baire-generic asymptotic behavior of dilation orbits on spaces of invariant measures, completing the program of Badea-Grivaux (Discrete Analysis 2024) by removing all restrictions on the sequence and on the matrices. Concretely, with $G_{p,(c_n)}$ and $G_{A,B}$ as defined in the Background, the averaged analogues $G''$ defined below, and the ambient spaces $(\mathcal{P}_p(\mathbb{T}),w^*)$ and $(\mathcal{P}_A(\mathbb{T}^d),w^*)$, answer each of the following four questions of the source paper:

(1) Question 5.5. For every integer $p\ge 2$ and every strictly increasing sequence of integers $(c_n)_{n\ge 0}$, is the set

$$G_{p,(c_n)}=\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;:\;T_{c_n}\mu\not\xrightarrow{\;w^*\;}\mathrm{Leb}\}$$

residual in $(\mathcal{P}_p(\mathbb{T}),w^*)$? (Known when $(c_n)$ satisfies the congruence criterion (H), e.g. $(q^n)$ and linearly recurrent sequences with coprime leading coefficient; open for arbitrary strictly increasing sequences.)

(2) Question 5.6. Let $p\ge 2$, and let $(c_n)$ be a strictly increasing sequence of integers. Determine whether the averaged-equidistribution set

$$G''_{p,(c_n)}=\Big\{\mu\in\mathcal{P}_{p,c}(\mathbb{T})\;:\;\frac{1}{N}\sum_{n=0}^{N-1}T_{c_n}\mu\xrightarrow{\;w^*\;}\mathrm{Leb}\Big\}$$

is residual in $(\mathcal{P}_p(\mathbb{T}),w^*)$. (The source records that "the question of the residuality remains widely open" and states that this is not known even for $(c_n)=(q^n)$ with $q$ multiplicatively independent from $p$. Note that the rigid choice $(c_n)=(p^n)$ is admissible and, since $T_{p^n}\mu=\mu$ for every $T_p$-invariant $\mu$, gives $G''_{p,(p^n)}=\{\mathrm{Leb}\}$, a singleton that is not dense and hence not residual in the Baire space $(\mathcal{P}_p(\mathbb{T}),w^*)$; the universal affirmative form therefore fails, and the question asks for a determination covering arbitrary admissible sequences rather than for such a single instance.)

(3) Question 5.8. For every $d\ge 2$ and all $A,B\in M_d(\mathbb{Z})$ (integer $d\times d$ matrices) with $\det A\ne 0$ and $\det B\ne 0$, is $G_{A,B}$ residual in $(\mathcal{P}_A(\mathbb{T}^d),w^*)$? Equivalently: are the hypotheses of the multidimensional theorem ($A$ diagonalizable over $\mathbb{C}$ without unit-modulus eigenvalues, and $\gcd(\det A,\det B)=1$ or $A$ triangular) genuinely necessary, or artifacts of the proof?

(4) Question 5.9. Under the same hypotheses as (3), determine, for an arbitrary such pair $(A,B)$, whether the averaged analogue (here $\mathcal{P}_{A,c}(\mathbb{T}^d)$ is the set of continuous, i.e. non-atomic, $T_A$-invariant measures on $\mathbb{T}^d$, and $\mathrm{Leb}_d$ is Lebesgue measure on $\mathbb{T}^d$)

$$G''_{A,B}=\Big\{\mu\in\mathcal{P}_{A,c}(\mathbb{T}^d)\;:\;\frac{1}{N}\sum_{n=0}^{N-1}T_{B^n}\mu\xrightarrow{\;w^*\;}\mathrm{Leb}_d\Big\}$$

is residual in $(\mathcal{P}_A(\mathbb{T}^d),w^*)$. (The source poses this question in analogy to Question 5.6. The rigid choice $B=A$ is admissible and, since $T_{A^n}\mu=\mu$ for every $T_A$-invariant $\mu$, gives $G''_{A,A}=\{\mathrm{Leb}_d\}$, a non-dense and hence non-residual singleton; as in (2), the question asks for a determination covering arbitrary admissible pairs rather than for a single instance.)

A complete answer resolves all four questions and thereby fixes, without any restriction on the sequence $(c_n)$ or the pair $(A,B)$, whether the generic continuous invariant measure fails to equidistribute along the full sequence while its Cesaro-averaged orbit equidistributes. For questions (1) and (3), each verdict remains possible: an affirmative answer requires constructing the dense $G_\delta$ witness sets (or an equivalent Baire argument) for the unrestricted class, while a negative answer requires an explicit strictly increasing sequence $(c_n)$, respectively an explicit matrix pair $(A,B)$ with nonzero determinants, for which the indicated set is not residual, together with a proof of non-residuality. For questions (2) and (4), by contrast, the rigid instances recorded above already refute the universal affirmative form, so a single explicit instance, in either direction, does not constitute a complete answer: what is required is a criterion deciding, for every strictly increasing sequence $(c_n)$, respectively every admissible pair $(A,B)$, whether $G''_{p,(c_n)}$, respectively $G''_{A,B}$, is residual — in particular settling the case $(c_n)=(q^n)$ with $q$ multiplicatively independent from $p$, which the source records as unknown. The neighboring conjecture (C7) of Lyons — $\mu$-almost-everywhere normality to base $q$ for every $\mu\in\mathcal{P}_{p,c}(\mathbb{T})$ — is a distinct famous open problem and is deliberately not part of this problem.

References

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