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On the $L^q$ dimension of stationary measures for M\"{o}bius iterated function systems

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Pith's one-line read The Lq spectrum of stationary measures for Möbius IFSs obeys a sharp dichotomy, and the second case is realized by explicit counterexamples.

desk verdict Counterexamples are solid and checkable; the main dichotomy is plausible but its 'only one of (I)/(II)' proof is missing from this preprint copy. read the letter →

arxiv 2501.13729 v1 pith:H43LRKKX submitted 2025-01-23 math.DS math.CAmath.CO

Shunsuke Usuki This is my paper · ORCID
classification math.DSmath.CAmath.CO MSC 28A8037C4537D35
keywords L^qdimensionstationarymeasureMöbiusiteratedfunctionsystemstronglyDiophantineconditionspectrummultifractalanalysisSL(2R)actionself-conformal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a complete dichotomy for the L^q spectrum of stationary measures of Möbius iterated function systems on the line that satisfy the strongly Diophantine condition. It shows that either the spectrum equals the candidate value $\min\{\tilde\tau(\nu,q), q-1\}$ for every $q>1$, where $\tilde\tau$ is the zero of a canonical pressure function, or it matches that candidate only up to a threshold $q_0$ and then becomes the linear function $\alpha q$ with $0<\alpha<1$. The paper further constructs explicit systems in which the second case occurs, and shows that this happens precisely when two distinct transformations share a fixed point. As a corollary, the natural extension of Shmerkin's L^q-dimension theorem from linear self-similar measures to Möbius IFSs fails in general.

What carries the argument

The proof is carried by the L^q norm flattening theorem, which says that under the assumption that the Legendre transform $\tau^*(\alpha)=\alpha q-\tau(q)$ is positive, convolving a flat measure on the group with the stationary measure strictly reduces its L^q norm. To prove flattening, the paper develops an L^q norm porosity lemma that linearizes the $\mathrm{SL}(2,\mathbb{R})$-action on $\mathbb{RP}^1$ at small scales, reduces to linear convolutions, and then applies Shmerkin's inverse theorem for L^q norms of linear convolutions (ultimately based on Bourgain's discretized projection theorem and the asymmetric Balog–Szemerédi–Gowers theorem). The positivity condition on the Legendre transform is exactly what distinguishes the two cases in the dichotomy: when it fails, the spectrum is linear from some point onward.

What would settle it

Compute, for one of the constructed examples (e.g., the family with $t=9^n$ and probabilities $p_0$ close to $1/2$ on the two maps sharing $0$), the L^q spectrum at a large $q$. If it does not become exactly linear in $q$ for all sufficiently large $q$, the dichotomy theorem fails; conversely, verifying the predicted linear tail would support it.

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Extended reading notes

Core claim

The central discovery is that the L^q spectrum $\tau(\nu,q)=(q-1)D(\nu,q)$ of a stationary measure for a uniformly hyperbolic, strongly Diophantine family in $\mathrm{SL}(2,\mathbb{R})$ has exactly one of two possible shapes: the 'expected' shape $\min\{\tilde\tau(\nu,q),q-1\}$ for all $q>1$, or the 'singular' shape in which the expected formula holds only for $1<q<q_0$ and $\tau(\nu,q)=\alpha q$ for $q\ge q_0$ with $0<\alpha<1$. The proof of the main theorem shows that the second case is realized when two different elements of the family share a common fixed point; the paper gives concrete parameters $t=9^n$ for which the semigroup is free, resolving a question of Solomyak.

Load-bearing premise

The equality in case (I) rests on the L^q norm flattening theorem, which in turn imports Shmerkin's inverse theorem for linear convolutions as a black box; if that inverse theorem or its additive-combinatorics ingredients had a gap, the equality $\tau=\min\{\tilde\tau,q-1\}$ would lack support.

Editorial extensions

If this is right

  • If correct, the theorem gives the complete L^q-dimension description for every stationary measure of a strongly Diophantine Möbius IFS: its spectrum is either the pressure candidate or a linear function beyond a critical exponent.
  • The dichotomy automatically rules out any smooth intermediate behavior: the spectrum cannot, for example, have a strictly concave piece after $q_0$.
  • The explicit counterexamples show that the strongly Diophantine and uniform hyperbolicity conditions do not suffice to make Shmerkin's linear theorem hold in the nonlinear setting; the obstruction is a common fixed point.
  • The L^q norm flattening theorem and porosity lemma provide a new route to multifractal information for non-conformal-like actions, and the paper's Proposition 1.10 shows that in the singular case there is a dense set of points of pointwise dimension $\alpha$ with Hausdorff dimension $0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One could test whether the dichotomy persists for general analytic IFSs on the line, as the author suggests in a remark; the shared-fixed-point mechanism would still be present, but the linearization tools might need replacement.
  • The examples with $t=9^n$ suggest a broader family: any algebraic parameter making the semigroup free should give a strongly Diophantine system, so the second case may be generic among systems with a common fixed point rather than exceptional.
  • The theorem has implications for the multifractal formalism: in case (II), the Legendre transform vanishes on $[\alpha,1]$, so the usual multifractal spectrum would be trivial beyond the linear phase; understanding the pointwise dimension distribution at the critical exponent $\alpha$ would be the next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper studies the L^q spectrum τ(ν,q) of stationary measures for Möbius IFSs, formulated as uniformly hyperbolic, strongly Diophantine finite families in SL2(R) acting on RP1. The central result (Theorem 1.8) is a claimed dichotomy: either τ(q)=min{τ̃(q),q−1} for every q>1, where τ̃(q) is the zero of the canonical pressure function, or there exist q0>1 and 0<α<1 such that the formula holds for 1<q<q0 and τ(q)=αq for q≥q0. The paper also proves (Theorem 1.7) that the natural formula fails when two distinct generators share a fixed point, and it constructs explicit examples (t=9n) answering Solomyak's question. The proof combines an Lq-norm porosity lemma, an Lq-norm flattening theorem obtained by linearizing the SL2(R)-action and invoking Shmerkin's inverse theorem for linear convolutions, and thermodynamic-formalism arguments. The copy under review is complete through Section 6.2 but breaks off in the middle of Section 6.3 during Lemma 6.7; consequently Proposition 1.5 and the 'only one of them' clause of Theorem 1.8 are not proved in this copy.

Significance. The dichotomy, if fully proved, would be a significant advance: it would be the first Lq-spectrum result of this type for non-linear IFSs, it shows that the natural extension of Shmerkin's linear-IFS theorem fails for Möbius systems, and it identifies a concrete mechanism (shared fixed points) through a rigorous elementary example. The counterexample section (Theorem 3.1) is correct as far as I verified: the mod-4 freeness argument and the pressure estimate are sound. The paper also deserves credit for stating explicitly the heavy black-box input (Shmerkin's inverse theorem, Theorem 5.4, with Bourgain's discretized projection and the asymmetric Balog–Szemerédi–Gowers theorem) on which the case (I) equality depends, and for the detailed proof of the porosity and flattening theorems. The main risk is completeness: the headline dichotomy is not established in the reviewed copy because Section 6.3 is missing.

major comments (2)
  1. [6.3, Theorem 1.8] The manuscript breaks off mid-sentence in the statement of Lemma 6.7 ('The function ψ : I^N → R is Hölder c…'). Section 6.3 is announced as the proof of Proposition 1.5 (analyticity of τ̃) and of the mutual exclusivity of cases (I) and (II). As a result, the 'only one of them' clause in Theorem 1.8 is unproved in the reviewed copy, and the theorem is not established as stated.
  2. [6.2 around (209)] The proof of the essential part of Theorem 1.8 extends the dichotomy from the dense set S of differentiability points to all q>1 by continuity, explicitly citing Proposition 1.5 for the continuity of τ̃(q). Since Proposition 1.5 is deferred to the missing Section 6.3, the conclusion τ(q)=min{τ̃(q),q−1} for all q>1 (case (I)), and the analogous extension in case (II), are not justified in this copy. This is load-bearing: without continuity of τ̃, the dichotomy is established only on the dense set S.
minor comments (4)
  1. [Lemma 6.6 proof] In the proof of Lemma 6.6, the line 'τ*(q1) = α1 − α0 = 0' should read '(α1 − α0)q0 = 0' (from which α1 = α0 and τ*(α1) = 0 follow); as printed, τ* is evaluated at a q-argument, which is a type error.
  2. [3.2, after (51)] In the display following (51), the intermediate product is written as (a 0; b 1), but all factors in (51) have zero first column, so the first column should be (0, b); the conclusion φ(V) = (0 0; 0 1) and φ(W) = (0 0; 1 1) is unaffected.
  3. [1.3, Theorem 1.7] In Theorem 1.7, the phrase 'sufficiently close to 1/2 in terms of A' should also ensure p0 > max_{j≠i0,j0} p_j, since the pressure estimate uses max_i p_i = p0; otherwise this condition remains implicit.
  4. [3.3, Proposition 1.10(iii)] The equality between the two counting sets in (iii) does not follow from the displayed upper bound alone; it also uses the lower bound τ(q)=αq (i.e., Σ ν(I)^q ≥ 2^{-(αq+ε)m}) to exclude intervals with mass above 2^{-(α−ε/q)m}. Adding one sentence would make the argument complete.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main dichotomy is derived from a black-box inverse theorem and direct pressure estimates; the truncated Section 6.3 is a proof gap, not circularity.

full rationale

No circular step is found. The main dichotomy is not circular: \tilde{\tau}(q) is defined as the unique zero of the pressure function \Psi_q, and the easy bound \tau\le\min\{\tilde{\tau},q-1\} is proved directly via the stopping-word identity in Lemma 6.3, not assumed. Equality in case (I) follows from Proposition 6.1, whose proof uses the L^q-norm flattening theorem (Theorem 1.12) under the explicit hypothesis \tau^*(\alpha)>0; this hypothesis concerns the measure's own spectrum and is exactly what fails in case (II). Theorem 1.12 is proved from the porosity lemma (Lemma 1.13) and Shmerkin's inverse theorem for linear convolutions (Theorem 5.4), imported as an external black box from [Shm19]; it is not a self-citation and its use is not circular. The examples in Section 3 are freeness checks for the semigroup {A,B,C_t} (Theorem 3.1), independent of the spectrum analysis. No parameter is fitted and then renamed as a prediction: the quantity \alpha in case (II) is determined structurally from concavity once the Legendre transform vanishes (Lemma 6.6). The only notable issue is completeness, not circularity: Proposition 1.5 (analyticity/continuity of \tilde{\tau}) is invoked in Section 6.2, but its proof in Section 6.3 cuts off mid-sentence in Lemma 6.7, so the continuity extension and the mutual-exclusion clause are not fully verified in the reviewed copy. This should be weighed as a proof gap or correctness risk, not as a circular reduction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No fitted constants enter the main dichotomy: τ̃(q) is the zero of the canonical pressure function, and q0 and α are derived from the measure's own spectrum. The only hand-chosen parameters appear in the counterexample construction (p0 near 1/2, t = 9n). The paper imports two deep external theorems as black boxes: Shmerkin's inverse theorem for Lq norms of linear convolutions (Theorem 5.4) and [HS17, Lemma 6.1], plus its own Proposition 1.5 (analyticity of τ̃), whose proof in Section 6.3 was only partially visible in the review copy. No new geometric or physical entities are postulated.

free parameters (2)
  • p0 (counterexample weight) = 0 < p0 < 1/2, chosen sufficiently close to 1/2 in terms of A
    In Theorem 1.7, the weights of the two maps sharing a fixed point are set to p0 close to 1/2 to make log(2p0) small enough that −(log 2p0)/(2 log r) < 1 and below τ̃(q)/q for large q; this hand-chosen parameter drives the counterexample (Section 3.1, inequality (48)).
  • t = 9n (example parameter) = t = 9n, n ∈ N
    In Section 3.2, the family {A, B, C_t} is strongly Diophantine precisely for t = 9n, where the mod-4 argument proves the semigroup is free. The value is chosen by hand to clear denominators in the conjugated matrices.
assumptions (5)
  • domain assumption Strongly Diophantine condition on the matrix family
    Definition 1.2; needed for Lemma 6.5 (effective separation of stopping words), which converts the flattening theorem into the equality τ(q) = τ̃(q) in case (I). The author notes in Section 1.2 that 'along a subsequence' would be more natural but is not treated.
  • domain assumption Uniform hyperbolicity of the family in SL(2,R)
    Definition 1.4; ensures a contracting action on RP1, existence of the attractor, coding map, stationary measure, and well-defined pressure function Ψ_q (Section 2.2).
  • standard math Shmerkin's inverse theorem for Lq norms of linear convolutions
    Theorem 5.4 is imported verbatim from [Shm19, Theorem 2.1] and is the engine of the flattening theorem's contradiction argument in Section 5.4. The paper does not re-prove it; it relies on Bourgain's discretized projection theorem and the asymmetric Balog-Szemeredi-Gowers theorem.
  • standard math Freeness implies strong Diophantine [HS17, Lemma 6.1]
    Section 3.2 uses this cited lemma to conclude that the free semigroup {A, B, C_t} for t = 9n gives a strongly Diophantine family.
  • standard math Existence and uniqueness of the stationary (Furstenberg) measure
    Section 2.3; standard for contracting IFSs, proved in outline using the coding map and stopping words.

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Pith. "Pith review of On the $L^q$ dimension of stationary measures for M\"{o}bius iterated function systems." pith.science (2026). https://pith.science/paper/H43LRKKX

@misc{pith2026250113729,
  author       = {Pith},
  title        = {Pith review of: On the $L^q$ dimension of stationary measures for M\"obius iterated function systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H43LRKKX}},
  note         = {Machine review of arXiv:2501.13729}
}
abstract

We study the $L^q$ dimension $D(\nu,q)\ (q>1)$ of stationary measures $\nu$ for M\"{o}bius iterated function systems on $\mathbb{R}$ satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem 6.6]{Shm19}. As the result, we show that there is the dichotomy: the $L^q$ spectrum $\tau(\nu,q)=(q-1)D(\nu,q)$ is equal to the desired value $\min\{\widetilde{\tau}(\nu,q),q-1\}$ for any $q>1$, where $\widetilde{\tau}(\nu,q)$ is the zero of the canonical pressure function, or there exist $q_0>1$ and $0<\alpha<1$ such that $\tau(\nu,q)=\min\{\widetilde{\tau}(\nu,q),q-1\}$ for $1<q<q_0$ and $\tau(\nu,q)=\alpha q$ for $q\geq q_0$. In addition, we give examples of M\"{o}bius iterated function systems which show the latter case by giving an affirmative answer to Solomyak's question \cite[Question 2]{Sol24}.

Figures

Figures reproduced from arXiv: 2501.13729 by the authors.

Figure 1
Figure 1. x1, x2, x3 mapped by k and a We can see that d(RP1 ) 3 (akx, akhx) = d(RP1 ) 3 (akx, akhk−1 a −1 · akx), akx ∈ (RP1 ) 3 is η0-separated. (21) Furthermore, by (19) and h ∈ BG ρε (1G), we have kakhk−1 a −1 − 1Gk ≤ kakka −1 kkh − 1Gk = e 2t kh − 1Gk ≤ Oη0 (ε −1 dG(h, 1G)) ≤ Oη0 (ε −1 ρε), (22) where we have used the fact that the norm metric on G and the metric dG by the left-invariant Riemannian metric of G is bi-Lips… view at source ↗

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