Dekking and Don (2010) settled the Palis conjecture for correlated fractal percolation: the algebraic difference of two independent copies contains an interval almost surely if and only if the sum of their Hausdorff dimensions exceeds one. Their machinery — a two-type branching process of left and right triangles whose expectation matrices M(0),...,M(M−1) encode the geometry of the difference, together with the distributed growth condition (DGC), a structural hypothesis on the pair of joint survival distributions — also yields a spectral characterization: for identical survival laws satisfying the DGC, interval presence is decided by whether the lower spectral radius ρ(Σ_M) of the matrix family exceeds or falls below one. Two regimes of this characterization remain open: the asymmetric case of two different survival laws with unequal marginals, for which no sharp criterion is known; and the critical regime ρ(Σ_M) = 1, together with the conjecture that at the critical dimension dimH F1 + dimH F2 = 1 the difference contains no interval except for deterministic sets. The problem asks to complete the spectral characterization in these remaining regimes: to extend the dichotomy to asymmetric pairs under the distributed growth condition, and to decide interval presence at criticality — by proof, or by explicit counterexamples where the dichotomy fails.

Background

The algebraic difference of two Cantor sets on the line, C2−C1={y−x:x∈C1,y∈C2}C_2 − C_1 = \{y − x : x ∈ C_1, y ∈ C_2\}, measures how two fractal sets interpenetrate under translation. It comes from dynamics: at homoclinic bifurcations of surface diffeomorphisms, the relative position of the stable and unstable Cantor sets controls the creation of infinitely many attractors, and these phenomena translate into the question of when such a difference contains an interval (Palis and Takens 1993). The relevant size is the Hausdorff dimension dimHdimH, the standard fractal dimension; the Palis conjecture predicts that generically C2−C1C_2 − C_1 contains an interval whenever dimHC1+dimHC2>1dimH C_1 + dimH C_2 > 1. For generic pairs of dynamically defined non-linear Cantor sets this was proved by Moreira and Yoccoz (2001); for generic linear Cantor sets the conjecture remains open.

This problem concerns the probabilistic counterpart, where the Cantor sets are random and the question sharpens to an exact criterion. An M-adic random Cantor set is generated as follows: divide [0,1][0,1] into M equal subintervals indexed by the alphabet A={0,…,M−1}A = \{0, …, M−1\} (read as a cyclic group, so indices are modulo M); at every node of the construction tree, draw the set of retained subintervals from a fixed probability measure µµ on the subsets of A — the joint survival distribution — independently at all nodes; iterate inside retained subintervals. The marginals are pi=∑X∋iµ(X)p_i = ∑_{X ∋ i} µ(X), the probability that subinterval i survives one step. If µ retains subintervals independently one obtains classical Mandelbrot (fractal) percolation; if µ is uniform on the m-element subsets of A one obtains correlated fractal percolation, also called (m, M, p)-percolation, where retention of sibling subintervals is dependent. In these level-independent models dimHFdimH F is almost surely constant given non-extinction (for uniform marginals p it equals 1+log⁡p/log⁡M1 + \log p / \log M). The Palis question for two independent such sets F1,F2F_1, F_2 with joint survival distributions µ and λ is: when does F1−F2F_1 − F_2 contain an interval almost surely? Dekking and Simon (2008) gave the first sharp conditions, proving in particular that for one-dimensional homogeneous Mandelbrot percolation a dimension sum above one implies an interval almost surely; Dekking, Simon and Székely (2011) proved the analogous statement for Larsson's random family; and Dekking and Don (2010) — the source of this problem — completely classified correlated fractal percolation, including the critical case: for (m, M, p)-percolation, F1−F2F_1 − F_2 contains an interval almost surely if and only if dimHF1+dimHF2>1dimH F_1 + dimH F_2 > 1.

The engine of these results is a branching-process description of the difference introduced by Dekking and Kuijvenhoven (2011). The difference F1−F2F_1 − F_2 is the image of the square Λ=F1×F2Λ = F_1 × F_2 under φ(x,y)=y−xφ(x, y) = y − x; after a 45-degree rotation, an interval appears in the difference precisely when, along vertical columns indexed by a shift k∈Ak ∈ A, the left and right triangles (the halves of the M-adic squares of Λ cut by the projection direction) proliferate. The triangle counts form a two-type branching process in a varying environment — a branching process whose reproduction law depends on the column shift, the two types being left and right triangles — with 2×2 expectation matrices M(k)M(k), k∈Ak ∈ A, whose entries are explicit expressions in the marginals (pi)(p_i) and (qi)(q_i): M(k)=(mk+1−Mmk−Mmk+1mk)M(k) = \begin{pmatrix} m_{k+1−M} & m_{k−M} \\ m_{k+1} & m_k \end{pmatrix}, where me=∑i−j=e, i,j∈Apiqjm_e = ∑_{i−j=e,\ i,j ∈ A} p_i q_j for e∈{−M,…,M}e ∈ \{−M, …, M\} and a sum over an empty index set equals 0 (this is the matrix form of Dekking–Kuijvenhoven, their Eq. (24), with the entries as in their Eq. (27) in the symmetric case µ=λµ = λ). Write ΣM={M(0),…,M(M−1)}Σ_M = \{M(0), …, M(M−1)\} and let the lower spectral radius (also called the joint spectral subradius) of a bounded set of matrices be ρ(Σ)=liminfn→∞min∣∣A1⋯An∣∣1/nρ(Σ) = liminf_{n→∞} min ||A_1 ⋯ A_n||^{1/n}, the minimum over all length-n products from Σ: the slowest exponential growth rate of matrix products, i.e. the triangle growth rate along the most favorable column sequences. First-order information is carried by the cyclic correlation coefficients γk=∑iqipi+kγ_k = ∑_i q_i p_{i+k}, the expected overlap of the two sets at shift k; these are also the column sums of the expectation matrices, [1 1] M(k)=[γk+1 γk][1\ 1]\,M(k) = [γ_{k+1}\ γ_k] (Dekking–Kuijvenhoven, Lemma 3.1).

Two structural hypotheses on the pair (µ,λ)(µ, λ) organize the known theorems. The joint survival condition (JSC) of Dekking–Kuijvenhoven asks that µ assign positive probability to the union of its marginal support; their Theorem 6.1 (Theorem 7.1 in the published version) proves, for pairs satisfying the JSC, having equal marginal probabilities, and leading to an irreducible ΣMΣ_M (the matrices admit no common invariant coordinate splitting, so the growth process cannot decouple), the sharp dichotomy: ρ(ΣM)>1ρ(Σ_M) > 1 implies F1−F2F_1 − F_2 contains an interval almost surely on the event that it is non-empty, and ρ(ΣM)<1ρ(Σ_M) < 1 implies it contains no interval almost surely. The distributed growth condition (DGC) of Dekking–Don asks that for every k∈Ak ∈ A there exist subsets Xk,Yk⊆AX_k, Y_k ⊆ A with µ(Xk)>0µ(X_k) > 0 and λ(Yk)>0λ(Y_k) > 0 such that every shifted overlap γe(Xk,Yk)=∑i1Yk(i)1Xk(i+e)γ_e(X_k, Y_k) = ∑_i 1_{Y_k}(i) 1_{X_k}(i+e) is at least 1 and the overlaps at shifts k and k+1 are at least 2; the DGC holds whenever the JSC holds with all γk>1γ_k > 1 (so it strictly generalizes the JSC), and under it the same first-order dichotomy holds: all γk>1γ_k > 1 implies an interval, while two consecutive γkγ_k below 1 imply no interval. An example of Dekking and Simon (M = 5, marginals (1,0,p,0,1)(1, 0, p, 0, 1), for which γ=(2+p2,1,2p,2p,1)γ = (2 + p², 1, 2p, 2p, 1) and ΣMΣ_M is reducible with ρ(ΣM)≥√(2p)>1ρ(Σ_M) ≥ √(2p) > 1 for p>1/2p > 1/2, yet F1−F2F_1 − F_2 contains no interval almost surely for every p<1p < 1) shows that some structural hypothesis is genuinely necessary. For identical survival laws (µ=λµ = λ) satisfying the DGC, Dekking–Don's Theorem 4 transfers the spectral characterization, irreducibility then following from the symmetry. What remains open — the subject of this problem — is everything outside that symmetric case: unequal marginals, and the critical value ρ(ΣM)=1ρ(Σ_M) = 1 (excluded from both branches of every theorem above).

Problem Statement

Let F1F_1 and F2F_2 be independent M-adic random Cantor sets on the line, generated by joint survival distributions µ and λ (probability measures on the subsets of the alphabet A={0,…,M−1}A = \{0, …, M−1\}). Let (pi)(p_i) and (qi)(q_i) be their marginals, let M(0),…,M(M−1)M(0), …, M(M−1) be the 2×2 expectation matrices of the associated two-type left/right-triangle branching process in varying environment, ΣM={M(0),…,M(M−1)}Σ_M = \{M(0), …, M(M−1)\}, and let ρ(ΣM)ρ(Σ_M) be the lower spectral radius of ΣMΣ_M. Complete the spectral characterization of almost-sure interval presence in the algebraic difference F1−F2F_1 − F_2, resolving the following two linked regimes:

(1) Asymmetric case. Determine whether the dichotomy — ρ(ΣM)>1ρ(Σ_M) > 1 implies F1−F2F_1 − F_2 contains an interval almost surely on {F1−F2≠∅}\{F_1 − F_2 ≠ ∅\}, and ρ(ΣM)<1ρ(Σ_M) < 1 implies F1−F2F_1 − F_2 contains no interval almost surely — extends to pairs with distinct joint survival distributions and unequal marginals under the distributed growth condition. Prove the extension, or exhibit an explicit asymmetric pair (µ,λ)(µ, λ) for which the dichotomy fails.

(2) Critical regime. Decide almost-sure interval presence when ρ(ΣM)=1ρ(Σ_M) = 1. A separate open conjecture of Dekking and Don (2010, final remarks of the arXiv version 0910.5865) concerns the dimensional rather than the spectral critical locus: at the critical dimension dimHF1+dimHF2=1dimH F_1 + dimH F_2 = 1 the difference F1−F2F_1 − F_2 contains no interval almost surely, except for deterministic sets (survival laws concentrated on a single retained subset). Prove or refute this conjecture, and decide the regime ρ(ΣM)=1ρ(Σ_M) = 1; the two critical loci are distinct, and no inclusion between them is asserted or presupposed.

References

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