In the proof of the large-dimension case of Palis's conjecture on the arithmetic difference of regular Cantor sets, Moreira and Yoccoz introduced a renormalization framework whose central output is a sufficient criterion (containment of a relative configuration of limit geometries in a recurrent compact set) for two dynamically defined Cantor sets to intersect stably. In a remark in Section 2.2 of that paper the authors note a second, a priori weaker way to define stable intersection for a configuration of limit geometries: persistence of the intersection under small perturbations of the two expansive maps defining the Cantor sets together with small perturbations of the configuration inside the space of affine configurations of limit geometries, instead of inside the much larger space of smooth configurations used in the paper's definition. They state that they do not know of any example that is stably intersecting in the weaker sense but not in the stronger sense. The question recorded here, left open by the source paper and untouched by the subsequent literature, is whether the two notions coincide: prove that every weakly stably intersecting configuration of limit geometries is strongly stably intersecting, or construct expansive maps and a configuration of limit geometries separating the two notions.

Background

A regular (dynamically defined) Cantor set is a compact, totally disconnected subset of the real line obtained as the maximal invariant set of a smooth expanding Markov map. Moreira and Yoccoz axiomatize the dynamics as follows. Fix a finite alphabet A\mathbb{A} and a set of allowed transitions B⊂A×A\mathcal{B}\subset\mathbb{A}\times\mathbb{A} whose associated subshift of finite type Σ\Sigma (the set of doubly infinite sequences of letters of A\mathbb{A} in which every consecutive pair is an allowed transition) is topologically mixing (any two letters of A\mathbb{A} can be joined by an admissible finite word). An expansive map of type Σ\Sigma is a map gg whose domain is the disjoint union of compact intervals I(a,b)I(a,b), (a,b)∈B(a,b)\in\mathcal{B}, with I(a,b)⊂I(a):=[0,1]×{a}I(a,b)\subset I(a):=[0,1]\times\{a\}, such that each restriction g∣I(a,b):I(a,b)→I(b)g|_{I(a,b)}:I(a,b)\to I(b) is a CrC^r diffeomorphism onto I(b)I(b) with ∣Dg∣>1|Dg|>1 on I(a,b)I(a,b); its maximal invariant set KK is a regular Cantor set, and gg is conjugate on KK to the one-sided shift on the space Σ+\Sigma^+ of right-infinite admissible sequences. For r>1r>1 (including r=∞r=\infty) the set of CrC^r expansive maps of type Σ\Sigma is denoted ΩΣr\Omega_\Sigma^r, and ΩΣ=⋃r>1ΩΣr\Omega_\Sigma=\bigcup_{r>1}\Omega_\Sigma^r carries the inductive-limit topology (a sequence converges when it is eventually defined in a common class CrC^r and converges in that class). The Hausdorff dimension HD(K)HD(K), the standard fractal dimension defined through covers by sets of small diameter, is a continuous and nontrivial invariant of gg (it is continuous even in the C1C^1 topology).

Intersections of pairs of regular Cantor sets control two classical phenomena. Newhouse introduced the thickness τ(K)\tau(K) of a regular Cantor set, a quantitative measure, at all scales, of how the length of a bridge of KK (the part of KK's convex hull adjacent to a gap) compares to the length of the gap; he proved the gap lemma (if τ(K)⋅τ(K′)>1\tau(K)\cdot\tau(K')>1 and the two convex hulls are interleaved, in the sense that neither lies inside a gap of the other, then KK and K′K' intersect, robustly under C1+αC^{1+\alpha}-small perturbations, since thickness is continuous in the C1+αC^{1+\alpha} topology), and used it to produce wild hyperbolic sets and persistent homoclinic tangencies (points where the stable and unstable manifolds of a periodic orbit meet tangentially) near surface diffeomorphisms with a homoclinic tangency (the Newhouse phenomenon). A classical fact, used by Palis and Takens in their study of homoclinic bifurcations, is that when HD(K)+HD(K′)<1HD(K)+HD(K')<1 the arithmetic difference K−K′={x−y:x∈K, y∈K′}K-K'=\{x-y:x\in K,\,y\in K'\} has Lebesgue measure zero, since its Hausdorff dimension is at most HD(K)+HD(K′)HD(K)+HD(K'). Palis conjectured, in the form proved in the source paper (see its introduction), that generically the arithmetic difference of two regular Cantor sets either has measure zero or contains an interval. Moreira introduced the modern notion of stable intersection of two regular Cantor sets, in which KK and K′K' intersect and every pair of Cantor sets defined by sufficiently small perturbations of the defining maps (in the same Markov setting) still intersects, and connected it to positive density of stable tangencies in generic one-parameter families of homoclinic bifurcations.

Moreira and Yoccoz (2001) proved the large-dimension case of Palis's conjecture: there is an open and dense set of pairs of C∞C^\infty expansive maps with HD(K)+HD(K′)>1HD(K)+HD(K')>1 such that, for each pair in this set, the set of translations tt for which KK and K′+tK'+t intersect stably is dense in the arithmetic difference K−K′K-K', and the complement of this stable-intersection set has Hausdorff dimension less than one. The proof rests on a renormalization scheme for configurations. A piece of KK associated to a finite admissible word a‾\underline a is the Cantor subset K(a‾)=K∩I(a‾)K(\underline a)=K\cap I(\underline a) obtained by following the branches of gg along a‾\underline a; a configuration is a Cr′C^{r'}-embedding hh (with r′>1r'>1) of one of the intervals I(a)I(a), and the space of all configurations is P=⋃a∈AP(a)\mathcal{P}=\bigcup_{a\in\mathbb{A}}\mathcal{P}(a), where P(a)=⋃r>1Pr(a)\mathcal{P}(a)=\bigcup_{r>1}\mathcal{P}^r(a) is glued from the spaces of CrC^r embeddings of I(a)I(a) into R\mathbb{R} by the inductive-limit topology. Renormalization operators precompose a configuration with inverse branches of gg, zooming toward smaller scales. Along a one-sided past θ‾∈Σ−\underline\theta\in\Sigma^- (a left-infinite admissible sequence), iterated renormalization converges to a limit geometry kθ‾k^{\underline\theta}, a Cr′C^{r'} diffeomorphism of the interval I(θ0)I(\theta_0) onto itself that is the canonical limiting shape of the tiny pieces of KK along θ‾\underline\theta; the limit geometries of a fixed gg depend continuously on θ‾\underline\theta and form a compact family. The space A={(θ‾,A)}\mathcal{A}=\{(\underline\theta,A)\}, where θ‾∈Σ−\underline\theta\in\Sigma^- and AA is an affine embedding of the interval I(θ0)I(\theta_0), embeds in P\mathcal{P} through the canonical map (θ‾,A)↦A∘kθ‾(\underline\theta,A)\mapsto A\circ k^{\underline\theta}, so an element of A\mathcal{A} is an affine configuration of limit geometries. For a pair of expansive maps g∈ΩΣg\in\Omega_\Sigma, g′∈ΩΣ′g'\in\Omega_{\Sigma'} with Cantor sets KK, K′K', a smooth configuration is a pair (h,h′)∈P×P′(h,h')\in\mathcal{P}\times\mathcal{P}'; it is linked if the images of the two domain intervals intersect, intersecting if the images of the corresponding pieces of KK and K′K' intersect, and stably intersecting if it is still intersecting after any sufficiently small perturbation of (h,h′)(h,h') in P×P′\mathcal{P}\times\mathcal{P}' together with any sufficiently small perturbation of (g,g′)(g,g') in ΩΣ×ΩΣ′\Omega_\Sigma\times\Omega_{\Sigma'}: this is the strong notion. Quotienting A×A′\mathcal{A}\times\mathcal{A}' by the diagonal action of the affine group (composition on the left by affine maps of the line) gives the space C\mathcal{C} of relative configurations of limit geometries, and the engine of the proof is the recurrent compact criterion: any relative configuration (of limit geometries) contained in a recurrent compact subset of C\mathcal{C} is stably intersecting in the strong sense, where a compact set is recurrent when every one of its points is carried into its interior by some nontrivial composition of renormalization operators.

In the Remark closing Section 2.2, Moreira and Yoccoz observe that for a configuration of limit geometries ((θ‾,A),(θ‾′,A′))((\underline\theta,A),(\underline\theta',A')) one could also consider a weaker notion of stable intersection, obtained by perturbing gg, g′g' in ΩΣ×ΩΣ′\Omega_\Sigma\times\Omega_{\Sigma'} and perturbing (θ‾,A)(\underline\theta,A), (θ‾′,A′)(\underline\theta',A') in A×A′\mathcal{A}\times\mathcal{A}': the perturbed configurations are required to remain affine configurations of limit geometries, rather than arbitrary smooth configurations as in the strong notion. They state that they do not know of any example of expansive maps and configurations that are stably intersecting in the weaker sense but not in the stronger sense. Since A×A′\mathcal{A}\times\mathcal{A}' sits inside P×P′\mathcal{P}\times\mathcal{P}' through the canonical map, strong stable intersection implies weak stable intersection; the open content of the remark is the converse.

The framework has since been extended in several directions, all of which define and use only the strong notion: stable homoclinic tangencies of product-type hyperbolic sets (hyperbolic invariant sets locally presented as products of one-dimensional dynamics; Moreira-Yoccoz 2010), conformal regular Cantor sets in the complex plane and the complex version of Palis's conjecture (Araújo-Moreira 2023; Moreira-Zamudio Espinosa 2024; Araújo-Moreira-Zamudio Espinosa 2025), the geometry of horseshoes (hyperbolic invariant Cantor sets of a diffeomorphism, the higher-dimensional analogues of regular Cantor sets; Moreira-Silva 2012), blender-based stable intersections in higher dimension (a blender is a hyperbolic set whose invariant laminations robustly intersect a whole open family of submanifolds; Asaoka 2021), and a covering criterion generalizing the recurrent compact criterion to arbitrary dimension (Nassiri-Zareh Bidaki 2025, 2026). Moreira (2011) showed on a different axis that no C1C^1-stable intersections of regular Cantor sets exist at all. None of these works addresses the equivalence of the two notions.

Problem Statement

Let A,B,Σ\mathbb{A},\mathcal{B},\Sigma and A′,B′,Σ′\mathbb{A}',\mathcal{B}',\Sigma' be data of topologically mixing subshifts of finite type (finite alphabets with sets of allowed transitions as in the background), and let gg, g′g' be expansive maps (of class CrC^r, r>1r>1) of types Σ\Sigma, Σ′\Sigma', defining regular Cantor sets KK, K′K'. In the framework of Moreira and Yoccoz (Annals of Mathematics (2) 154 (2001), 45-96, Sections 1.5 and 2.1-2.2), denote by ΩΣ\Omega_\Sigma, ΩΣ′\Omega_{\Sigma'} the spaces of expansive maps (unions over r>1r>1 of the classes of CrC^r expansive maps, with the inductive-limit topology), by P\mathcal{P}, P′\mathcal{P}' the spaces of smooth configurations (P=⋃aP(a)\mathcal{P}=\bigcup_{a}\mathcal{P}(a), where P(a)\mathcal{P}(a) is the union over r′>1r'>1 of the spaces of Cr′C^{r'} embeddings of the interval I(a)=[0,1]×{a}I(a)=[0,1]\times\{a\} into R\mathbb{R}, with the inductive-limit topology), by Σ−\Sigma^-, Σ′−\Sigma'^- the spaces of left-infinite admissible sequences, by kθ‾k^{\underline\theta}, k′θ‾′k'^{\underline\theta'} (θ‾∈Σ−\underline\theta\in\Sigma^-, θ‾′∈Σ′−\underline\theta'\in\Sigma'^-) the limit geometries of gg, g′g' (Cr′C^{r'} diffeomorphisms of I(θ0)I(\theta_0), I′(θ0′)I'(\theta'_0) onto themselves), and by A×A′\mathcal{A}\times\mathcal{A}' the space of pairs ((θ‾,A),(θ‾′,A′))((\underline\theta,A),(\underline\theta',A')) with AA, A′A' affine embeddings of I(θ0)I(\theta_0), I′(θ0′)I'(\theta'_0), topologized as bundles over Σ−\Sigma^-, Σ′−\Sigma'^- (two points with the same present letter are close when the ultrametric distance between the pasts, d(θ‾,θ~‾)=∣I(θ‾∧θ~‾)∣d(\underline\theta,\underline{\tilde\theta})=|I(\underline\theta\wedge\underline{\tilde\theta})|, is small and the affine maps are coefficientwise close), and mapped into P×P′\mathcal{P}\times\mathcal{P}' by the canonical map ((θ‾,A),(θ‾′,A′))↦(A∘kθ‾,A′∘k′θ‾′)((\underline\theta,A),(\underline\theta',A'))\mapsto(A\circ k^{\underline\theta},A'\circ k'^{\underline\theta'}).

A configuration of limit geometries ((θ‾,A),(θ‾′,A′))((\underline\theta,A),(\underline\theta',A')) is intersecting if the images of the corresponding Cantor pieces meet, that is, if (A∘kθ‾)(K∩I(θ0))∩(A′∘k′θ‾′)(K′∩I′(θ0′))≠∅(A\circ k^{\underline\theta})(K\cap I(\theta_0))\cap(A'\circ k'^{\underline\theta'})(K'\cap I'(\theta'_0))\neq\varnothing. It is stably intersecting in the weaker sense if there exist a neighborhood UU of (g,g′)(g,g') in ΩΣ×ΩΣ′\Omega_\Sigma\times\Omega_{\Sigma'} and a neighborhood WW of ((θ‾,A),(θ‾′,A′))((\underline\theta,A),(\underline\theta',A')) in A×A′\mathcal{A}\times\mathcal{A}' such that, for every (g~,g~′)∈U(\tilde g,\tilde g')\in U with Cantor sets K~\tilde K, K~′\tilde K' and limit geometries k~θ~‾\tilde k^{\underline{\tilde\theta}}, k~′θ~‾′\tilde k'^{\underline{\tilde\theta}'}, and for every ((θ~‾,A~),(θ~‾′,A~′))∈W((\underline{\tilde\theta},\tilde A),(\underline{\tilde\theta}',\tilde A'))\in W, one has (A~∘k~θ~‾)(K~∩I(θ~0))∩(A~′∘k~′θ~‾′)(K~′∩I′(θ~0′))≠∅(\tilde A\circ\tilde k^{\underline{\tilde\theta}})(\tilde K\cap I(\tilde\theta_0))\cap(\tilde A'\circ\tilde k'^{\underline{\tilde\theta}'})(\tilde K'\cap I'(\tilde\theta'_0))\neq\varnothing. It is stably intersecting in the stronger sense if there exist a neighborhood UU of (g,g′)(g,g') in ΩΣ×ΩΣ′\Omega_\Sigma\times\Omega_{\Sigma'} and a neighborhood VV of (A∘kθ‾,A′∘k′θ‾′)(A\circ k^{\underline\theta},A'\circ k'^{\underline\theta'}) in P×P′\mathcal{P}\times\mathcal{P}' such that, for every (g~,g~′)∈U(\tilde g,\tilde g')\in U with Cantor sets K~\tilde K, K~′\tilde K' and for every (h,h′)∈V(h,h')\in V, one has h(K~∩I(θ0))∩h′(K~′∩I′(θ0′))≠∅h(\tilde K\cap I(\theta_0))\cap h'(\tilde K'\cap I'(\theta'_0))\neq\varnothing. Since the canonical map embeds A×A′\mathcal{A}\times\mathcal{A}' in P×P′\mathcal{P}\times\mathcal{P}', the stronger notion implies the weaker. Decide whether the two notions coincide: either prove that every configuration of limit geometries of expansive maps that is stably intersecting in the weaker sense is stably intersecting in the stronger sense, or exhibit expansive maps gg, g′g' and a configuration ((θ‾,A),(θ‾′,A′))((\underline\theta,A),(\underline\theta',A')) which is stably intersecting in the weaker sense but not in the stronger sense.

References

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