Exceptions in the domain of generic absolute continuity of non-homogeneous self-similar measures

Non-homogeneous self-similar measures are generically absolute continuous in the domain of parameters for which the similarity dimension is larger than one, see \cite{[SSS]}. Using certain algebraic curves we construct here exceptional singular non-homogeneous self-similar measures in this domain.

💡 Research Summary

The paper investigates the absolute continuity of non‑homogeneous self‑similar measures μ_{p,β₁,β₂} generated by the two affine contractions T₁(x)=β₁x+β₁ and T₂(x)=β₂x−β₂ on ℝ, where β₁,β₂∈(0,1) and p∈(0,1) is the weight of T₁. It is known that when the similarity dimension
S_D(p,β₁,β₂)=−p log p−(1−p) log(1−p)−p log β₁−(1−p) log β₂
exceeds one, the measure is absolutely continuous for “almost all’’ choices of the parameters; this generic result follows from transversality techniques and was refined in recent works (e.g., Saglietti‑Shmerkin‑Solomyak, 2018).

The author shows that this generic picture is not complete: there exist explicit parameter pairs (β₁,β₂) for which S_D>1 yet the corresponding measure is singular (its Hausdorff dimension is strictly less than one). The construction relies on a family of algebraic curves C defined as follows. Fix an integer n≥3 and two distinct sign sequences s,t∈{−1,1}ⁿ with the same number of +1 entries. For each k≤n let #_k(s) be the number of +1’s among the first k entries of s and \tilde#_k(s)=k−#k(s). The curve c{s,t} is given by the equation

∑_{k=1}^n s_k x^{#_k(s)} y^{\tilde#k(s)} − ∑{k=1}^n t_k x^{#_k(t)} y^{\tilde#_k(t)}=0

in the (x,y)‑plane. The set C consists of all such curves for all admissible (s,t).

The main theorem (Theorem 1.2) states: if (β₁,β₂) lies on a curve from C and satisfies β₁+β₂>1 (i.e., it belongs to the open rectangle R={ (x,y)∈(0,1)² | x+y>1 }), then there exists a probability p such that S_D(p,β₁,β₂)>1 while the Hausdorff dimension of μ_{p,β₁,β₂} is strictly smaller than one. The proof proceeds in two steps. First, Proposition 2.1 shows that on any such curve the effective similarity dimension d_{SD}(p,β₁,β₂) of the reduced iterated function system (obtained by merging the two maps corresponding to s and t, which coincide on the curve) satisfies d_{SD}<S_D for every p. Consequently, d_{SD} provides an upper bound for the Hausdorff dimension of the measure. Second, Proposition 2.2 exploits the condition β₁+β₂>1: one can choose d>1 with β₁^{d}+β₂^{d}=1 and set p=β₁^{d}. Then S_D(p,β₁,β₂)=d>1, while continuity of the functions p↦S_D and p↦d_{SD} guarantees the existence of a p where d_{SD}<1<S_D.

Concrete examples are supplied. For n=5, the sequences s=(1,−1,−1,−1,1) and t=(−1,1,1,−1,−1) generate the curve

2x²y³ + x²y² − x²y − xy³ − xy² − 2xy + x + y = 0,

which indeed intersects R (see Figure 1). For n=6 the author lists four further (s,t) pairs and the corresponding algebraic equations, all of which have points in R.

To determine when a curve from C actually meets R, Proposition 3.1 gives a sufficient condition: if s starts with (1,−1), t starts with (−1,1), and the weighted sums ∑_{k=1}^n s_k #k(s) and ∑{k=1}^n t_k #_k(t) satisfy the strict inequality ∑ s_k #_k(s) > ∑ t_k #_k(t), then the curve does not intersect R. This criterion is used to verify that many potential (s,t) choices do not produce exceptions, while the previously mentioned examples do.

In summary, the paper demonstrates that the set of parameters for which non‑homogeneous self‑similar measures are singular is not negligible: it contains explicit algebraic curves within the region where the similarity dimension exceeds one. This refines the “almost everywhere’’ absolute continuity results and opens new questions about the size and structure of the exceptional set, as well as the possible classification of all curves in C that intersect the region R.