Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbb{R}$

Ariel Rapaport

Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbb{R}$

T0 review · 0 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that exponentially separated real-analytic iterated function systems without a common fixed point yield self-conformal measures whose dimension equals the entropy-to-Lyapunov ratio, capped at one.

desk verdict Extends Hochman's inverse-theorem approach to real analytic IFSs, with a genuinely new finite-dimensional reduction; the proof looks sound and the main theorem is a real advance, though it leans on imported results. read the letter →

arxiv 2412.16753 v2 pith:F64UMHZC submitted 2024-12-21 math.DS

Ariel Rapaport This is my paper · ORCID

classification math.DS MSC 28A8037C45

keywords self-conformalmeasuresexponentialseparationrealanalyticIFSdimensionofentropyincreaseLyapunovexponentiteratedfunctionsystemsfractalgeometry

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 that the natural entropy-to-Lyapunov dimension formula holds for every self-conformal measure generated by a real-analytic iterated function system on the interval, provided the maps are uniformly contracting, have no common fixed point, and are exponentially separated. Earlier theorems of this kind were confined to self-similar systems or to maps acting through a finite-dimensional Lie group. The paper replaces the infinite-dimensional space of all real-analytic maps by a finite-dimensional space of polynomials of bounded degree, where an entropy-increase theorem can be applied. If correct, the result gives the first confirmation of the expected dimension formula for a broad class of genuinely non-linear, infinite-dimensional analytic IFSs, and yields as corollaries the corresponding dimension formula for self-conformal sets and for almost every member of a one-parameter analytic family.

What carries the argument

The key machinery is the reduction of convolutions $\nu.\mu$, where $\nu$ is a measure on the infinite-dimensional space $A(I)$ of real-analytic maps, to convolutions $\nu'.\mu$ with $\nu'$ supported on the finite-dimensional space $P_k$ of polynomials of degree at most $k$. Taylor's theorem with Lagrange remainder, together with the bounded-distortion estimate (Lemma 2.1) bounding all higher derivatives of iterates $\varphi_u$ in terms of $|\varphi'_u(0)|$, makes the error uniform and exponentially small. On $P_k$, an entropy-increase theorem (Theorem 3.1) is proved along the lines of earlier work on convolutions of self-similar and self-affine measures with measures on affine maps: it uses the inverse theorem for entropy from [7] and a uniform-entropy-dimension statement for $C^1$-images of $\mu$. The contradiction argument also uses the conditional-entropy identity from [6], namely $\dim\mu = (H(p) - \Delta')/\chi$, which converts a dimension drop into the positivity of the conditional entropy $\Delta'$.

What would settle it

The claim would be refuted by exhibiting an exponentially separated real-analytic IFS on [0,1] with no common fixed point, a positive probability vector p, and a self-conformal measure μ with dim μ < min{1, H(p)/χ}. A concrete check is whether the identity from [6, Theorem 2.8] holds for any such system; if not, that is the point to test. A numerical computation of dim μ for a polynomial IFS at algebraic parameters would be a practical test.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for an analytic IFS $\Phi$ on $I=[0,1]$ with $0<|\varphi_i'(x)|<1$, no common fixed point, and exponential separation, every positive probability vector $p$ produces a self-conformal measure $\mu$ with $\dim\mu = \min\{1, H(p)/\chi(\Phi,p)\}$. The proof works by contradiction: assume a dimension deficit $\Delta = \min\{1,H/\chi\} - \dim\mu > 0$, then decompose the semigroup according to a cut set and Taylor-expand the maps to order $k$. The analyticity and bounded distortion give uniform control on higher derivatives, so convolutions with measures on real-analytic maps are well approximated by convolutions with measures on polynomials of degree at most $k$. For those polynomial convolutions, an entropy-increase theorem (Theorem 3.1) shows that the entropy of the convolved measure exceeds that of $\mu$ by a fixed amount, which contradicts the assumed dimension drop. The no-common-fixed-point condition guarantees $\mu$ is non-atomic and the attractor is not a point, which is necessary for the formula.

Load-bearing premise

The entire contradiction argument collapses if the imported identity dim μ=(H(p)−Δ′)/χ fails for some exponentially separated analytic IFS without a common fixed point, and the paper does not prove this identity from scratch.

Editorial extensions

If this is right

Theorem 1.2 gives the dimension formula for every exponentially separated real-analytic IFS without a common fixed point, going beyond any finite-dimensional acting group.
Corollary 1.3 computes the Hausdorff dimension of the self-conformal set as $\min\{1, s(\Phi)\}$, where $s(\Phi)$ is the conformal similarity dimension.
Corollary 1.4 shows that in a non-degenerate one-parameter analytic family of IFSs, all but a Hausdorff-dimension-zero set of parameters satisfy both the measure and set dimension formulas.
Whenever the dimension formula fails for an exponentially separated analytic IFS, the proof forces strong restrictions on the system and ties the size of the failure to the conditional entropy $\Delta'$.
The author states that the proof strategy is expected to transfer other recent results on stationary fractal measures to the real-analytic setting.

Reading between the lines

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

The Taylor-polynomial reduction suggests a general template: any entropy-increase statement for finite-dimensional parameter spaces can lift to infinite-dimensional analytic families whenever all higher derivatives are controlled by the first derivative; this could extend to random walks on diffeomorphism groups.
The analyticity assumption may be close to necessary: the proof fails for $C^\infty$ maps, and a $C^\infty$ exponentially separated system with unbounded derivative growth would be a natural test of how far the theorem can stretch.
The paper leaves open whether exponential separation is equivalent to freeness for polynomial IFSs with algebraic coefficients, because degrees of iterates grow exponentially; a proof or counterexample for quadratic maps would sharpen the exact-overlaps picture in the analytic case.
Because the bottleneck is the imported conditional-entropy identity, a promising route to non-analytic conformal IFSs is to prove $\dim\mu=(H(p)-\Delta')/\chi$ afresh under exponential separation alone.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline is that this paper delivers the first exponential-separation dimension formula for self-conformal measures outside finite-dimensional Lie group settings. Hochman covered self-similar maps; Hochman–Solomyak covered Möbius systems in SL(2,R). Rapaport gets all real analytic contractions with no common fixed point. That is a genuine extension, and the main novelty—reducing convolutions against the infinite-dimensional space of analytic maps to convolutions against bounded-degree polynomials—is clever and seems to work. The entropy increase argument for polynomial convolutions is carefully assembled, and the constant hierarchy M << k << ρ^{-1} << δ^{-1} << n is transparent enough to follow without getting lost.

I looked at the spots that usually kill this kind of paper. The imported identity from Feng–Hu (dim μ = (H(p) − Δ')/χ) is unconditional for C^{1+γ} conformal IFS, so using it to detect a dimension drop is legitimate, not circular. The exponential separation step at (4.5) is justified by the scaling argument in the stress-test note: separation at infinitely many scales gives a uniform constant at the intermediate scales actually needed. The Taylor remainder estimate in (4.14) checks out, and the polynomial map p is bounded with derivative bounded away from zero, matching the hypotheses of Theorem 3.1.

Soft spots are minor and mostly about delegation. Lemmas 3.8 and 3.9 are omitted with references to earlier work; they are plausibly routine, but a referee should verify them since they do real work in the entropy-increase proof. The paper is also honest about its limits: analyticity is used for derivative bounds and cannot be relaxed to C^∞ here, and the authors do not claim algebraic parameter systems are exponentially separated. None of this undermines the main theorem.

This is a paper for fractal geometry and dimension theory readers. It deserves a serious referee: the result is important within the area, the proof is inventive, and the exposition is unusually clear about the hierarchy of parameters. I would send it to review without hesitation.

Referee Report

0 major / 5 minor

Summary. This paper proves (Theorem 1.2) that for a real-analytic iterated function system Phi on [0,1] with 0<|phi_i'(x)|<1 for all i and x, with no common fixed point, and exponentially separated in the sense of Definition 1.1, every positive self-conformal measure mu satisfies dim mu = min{1,H(p)/chi}. The proof proceeds by contradiction: a positive dimension drop is extracted from the Feng–Hu identity dim mu = (H(p)-Delta')/chi, and the exponential separation is used to force, through a Taylor approximation of analytic semigroup elements, a large entropy increase for convolutions of mu with measures on spaces of polynomials. The entropy increase result (Theorem 3.1) is the technical core of the paper. The paper also derives the corresponding formula for self-conformal sets (Corollary 1.3) and a zero-dimensional exceptional set statement for one-parameter analytic families (Corollary 1.4).

Significance. If correct, Theorem 1.2 is a significant extension of Hochman's inverse-theorem approach beyond finite-dimensional parameter spaces: it appears to be the first exponential-separation dimension result for self-conformal measures whose IFS is not contained in a finite-dimensional Lie group. The reduction of convolutions with measures on the infinite-dimensional space of real analytic maps to convolutions with measures on finite-dimensional polynomial spaces is a genuine new idea and is likely to be reused in other stationary fractal settings. The proof is carefully organized, with a transparent hierarchy of parameters (M << k << rho^{-1} << delta^{-1} << n) and with external tools (Hochman's inverse theorem, the Feng–Hu dimension formula) imported as black boxes; I found no internal inconsistency or circular dependence on the main conclusion. The corollaries, especially the parametric family statement, increase the paper's utility.

minor comments (5)

[Section 4, Eq. (4.5)] The sentence 'By exponential separation and the choice of c, we may clearly assume that (4.5) holds' is too terse. Exponential separation gives a lower bound c^m only at infinitely many scales m, whereas the proof needs a uniform bound at level n+n' for the particular n chosen in the parameter hierarchy. The step is valid: one chooses m comparable to (1+alpha)n with alpha = Delta/(2 log |Lambda|) and replaces c by a fixed power c^{1+alpha}; please spell this out explicitly.
[Section 3.4, around Eq. (3.15)] The application of Lemma 3.9 to the component measures nu_{p,i} and theta_{x,i} suppresses the usual truncation of small i. For the lemma to apply, the component diameter 2^{-i} must be smaller than the delta supplied by Lemma 3.9; this holds for i >= i0, and the finitely many exceptional indices contribute an error O(i0/n) that is absorbed by taking n large in the hierarchy (3.14). A sentence making this truncation explicit would improve rigor.
[Section 3.3, Lemmas 3.8 and 3.9] Lemmas 3.8 and 3.9 are stated without proof and are said to follow from [10, Lemma 6.9] and [1, Lemma 4.2], respectively. Since Theorem 3.1 rests on these two lemmas, the paper should either include their proofs or state precisely how the referenced arguments adapt to the present notation and to the class of measures under consideration.
[Lemma 4.1 proof] The phrase 'basic basic properties of disintegrations' contains a duplicated word; also, the verification that the relevant properties hold for beta-almost every omega would read more cleanly if the two displayed limit statements were labelled.
[Corollary 1.3 proof] The verification that Phi_n inherits the hypotheses of Theorem 1.2 (no common fixed point and exponential separation) is only sketched as 'easy to see'. A short justification would be helpful, since Corollary 1.3 depends on applying the main theorem to Phi_n.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorem 1.2 is proved from Feng-Hu's unconditional dimension identity and Hochman's inverse theorem, not from its own conclusion.

full rationale

The central derivation is not circular. The contradiction argument in Theorem 1.2 imports the identity dim μ = (H(p) − Δ′)/χ from [6, Theorem 2.8] as an unconditional result for the present C^{1+γ} conformal setting; this identity is not the theorem's conclusion, and it is used only to convert the assumed dimension drop into Δ′ > 0. The entropy increase result (Theorem 3.1) is then proved from Hochman's inverse theorem, uniform entropy dimension estimates, and a polynomial approximation argument; none of these steps assumes the desired formula min{1, H(p)/χ}. The no-common-fixed-point assumption is used only to guarantee non-atomicity, not to encode the dimension formula. The only self-referential element is that Lemmas 3.8 and 3.9 are stated with proofs omitted and said to follow [10, Lemma 6.9] and [1, Lemma 4.2], both involving the author; these are auxiliary technical estimates within the entropy-increase proof, and the main claim does not reduce to them. Thus no load-bearing circular step is present; at most there are minor non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical parameters are fitted. The proof constants M, k, n, δ, ρ are selected according to the hierarchy in (4.4) to make inequalities true; they are not fitted to data and carry no independent scientific content. The only imported mathematical entities are standard ones: Taylor polynomials, dyadic partitions, and cited theorems from Hochman and Feng-Hu.

assumptions (4)

domain assumption The system {φ_i} is a real analytic IFS on I with 0<|φ_i'(x)|<1 for all i and x, and the maps have no common fixed point.
These are the standing assumptions of Theorem 1.2; they are used throughout, for example to ensure non-atomicity and bounded distortion.
domain assumption The maps extend holomorphically to a common complex neighbourhood I(ε) with uniform derivative bounds.
Used in Lemma 2.1 to bound higher derivatives of semigroup elements. It follows from finite analyticity and compactness, but the uniform constants are imported.
standard math Feng-Hu dimension identity dim μ = (H(p) - Δ')/χ holds for this class, together with the related convergence in (4.3).
Cited as [6, Theorem 2.8] and [6, Proposition 4.10]. This is load-bearing for converting the assumed dimension drop into a positive fiber entropy Δ'.
standard math Hochman's inverse theorem for entropy (Theorem 3.2 in the paper) is valid at the required scales.
Cited as [7, Theorem 2.8] via [1, Section 4.1]. It supplies the entropy increase for convolutions in R that drives Theorem 3.1.

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Pith. "Pith review of Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbb{R}$." pith.science (2026). https://pith.science/paper/F64UMHZC

@misc{pith2026241216753,
  author       = {Pith},
  title        = {Pith review of: Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbbR$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F64UMHZC}},
  note         = {Machine review of arXiv:2412.16753}
}

abstract

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$ consisting of real analytic contractions, let $p=(p_{i})_{i\in\Lambda}$ be a positive probability vector, and let $\mu$ be the associated self-conformal measure. Suppose that the maps in $\Phi$ do not have a common fixed point, $0<\left|\varphi_{i}'(x)\right|<1$ for $i\in\Lambda$ and $x\in I$, and $\Phi$ is exponentially separated. Under these assumptions, we prove that $\dim\mu=\min\left\{ 1,H(p)/\chi\right\} $, where $H(p)$ is the entropy of $p$ and $\chi$ is the Lyapunov exponent. The main novelty of our work lies in an argument that reduces convolutions of $\mu$ with measures on the (infinite-dimensional) space of real analytic maps to convolutions with measures on vector spaces of polynomials of bounded degree. The reason for this reduction is that, for the latter convolutions, we can establish an entropy increase result, which plays a crucial role in the proof. We believe that our proof strategy has the potential to extend other significant recent results in the dimension theory of stationary fractal measures to the real analytic setting.

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Works this paper leans on

19 extracted references · 17 canonical work pages · cited by 3 Pith papers

[1]

Bárány, M

B. Bárány, M. Hochman, and A. Rapaport. Hausdorﬀ dimensi on of planar self-aﬃne sets and measures. Invent. Math. , 216(3):601–659, 2019

work page 2019
[2]

Bárány, K

B. Bárány, K. Simon, and B. Solomyak. Self-similar and self-aﬃne sets and measures , volume 276 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, [2023] ©2023

work page 2023
[3]

R. Bowen. Equilibrium states and the ergodic theory of Anosov diﬀeomo rphisms, volume 470 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, revised edition, 2008. With a preface by David Ruelle

work page 2008
[4]

Einsiedler and T

M. Einsiedler and T. W ard. Ergodic theory with a view towards number theory , volume 259 of Graduate Texts in Mathematics . Springer-Verlag London, Ltd., London, 2011

work page 2011
[5]

Feng and Z

D.-J. Feng and Z. Feng. Dimension of homogeneous iterate d function systems with algebraic translations, 2024. arXiv:2405.03124

arXiv 2024
[6]

Feng and H

D.-J. Feng and H. Hu. Dimension theory of iterated functi on systems. Comm. Pure Appl. Math., 62(11):1435–1500, 2009

work page 2009
[7]

M. Hochman. On self-similar sets with overlaps and inver se theorems for entropy. Ann. of Math. (2) , 180(2):773–822, 2014

work page 2014
[8]

M. Hochman. Dimension theory of self-similar sets and me asures. In Proceedings of the Inter- national Congress of Mathematicians—Rio de Janeiro 2018. Vo l. III. Invited lectures , pages 1949–1972. W orld Sci. Publ., Hackensack, NJ, 2018

work page 2018

Show all 19 references

[9]

M. Hochman. On self-similar sets with overlaps and inver se theorems for entropy in Rd. Mem. Amer. Math. Soc. , 265(no. 1287), 2021

2021
[10]

Hochman and A

M. Hochman and A. Rapaport. Hausdorﬀ dimension of plana r self-aﬃne sets and measures with overlaps. J. Eur. Math. Soc. (JEMS) , 24(7):2361–2441, 2022. 22

2022
[11]

Hochman and B

M. Hochman and B. Solomyak. On the dimension of Furstenb erg measure for SL2(R) random matrix products. Invent. Math. , 210(3):815–875, 2017

2017
[12]

J. E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J. , 30(5):713–747, 1981

1981
[13]

Rapaport

A. Rapaport. Proof of the exact overlaps conjecture for systems with algebraic contractions. Ann. Sci. Éc. Norm. Supér. (4) , 55(5):1357–1377, 2022

2022
[14]

Rapaport and P

A. Rapaport and P. Varjú. Self-similar measures associ ated to a homogeneous system of three maps. Duke Math. J. , 173(3):513–602, 2024

2024
[15]

Simon, B

K. Simon, B. Solomyak, and M. Urbański. Invariant measu res for parabolic IFS with overlaps and random continued fractions. Trans. Amer. Math. Soc. , 353(12):5145–5164, 2001

2001
[16]

Solomyak and Y

B. Solomyak and Y. Takahashi. Diophantine property of m atrices and attractors of projective iterated function systems in RP1. Int. Math. Res. Not. IMRN , (16):12639–12669, 2021

2021
[17]

P. Varjú. On the dimension of Bernoulli convolutions fo r all transcendental parameters. Ann. of Math. (2) , 189(3):1001–1011, 2019

2019
[18]

P. Varjú. Self-similar sets and measures on the line. In ICM—International Congress of Mathematicians. Vol. 5. Sections 9–11 , pages 3610–3634. EMS Press, Berlin, [2023] ©2023

2023
[19]

L. S. Young. Dimension, entropy and Lyapunov exponents . Ergodic Theory Dynam. Systems , 2(1):109–124, 1982. Department of Mathematics, Technion, Haif a, Israel E-mail: arapaport@technion.ac.il 23

1982

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Bárány, M

B. Bárány, M. Hochman, and A. Rapaport. Hausdorﬀ dimensi on of planar self-aﬃne sets and measures. Invent. Math. , 216(3):601–659, 2019

work page 2019

[2] [2]

Bárány, K

B. Bárány, K. Simon, and B. Solomyak. Self-similar and self-aﬃne sets and measures , volume 276 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, [2023] ©2023

work page 2023

[3] [3]

R. Bowen. Equilibrium states and the ergodic theory of Anosov diﬀeomo rphisms, volume 470 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, revised edition, 2008. With a preface by David Ruelle

work page 2008

[4] [4]

Einsiedler and T

M. Einsiedler and T. W ard. Ergodic theory with a view towards number theory , volume 259 of Graduate Texts in Mathematics . Springer-Verlag London, Ltd., London, 2011

work page 2011

[5] [5]

Feng and Z

D.-J. Feng and Z. Feng. Dimension of homogeneous iterate d function systems with algebraic translations, 2024. arXiv:2405.03124

arXiv 2024

[6] [6]

Feng and H

D.-J. Feng and H. Hu. Dimension theory of iterated functi on systems. Comm. Pure Appl. Math., 62(11):1435–1500, 2009

work page 2009

[7] [7]

M. Hochman. On self-similar sets with overlaps and inver se theorems for entropy. Ann. of Math. (2) , 180(2):773–822, 2014

work page 2014

[8] [8]

M. Hochman. Dimension theory of self-similar sets and me asures. In Proceedings of the Inter- national Congress of Mathematicians—Rio de Janeiro 2018. Vo l. III. Invited lectures , pages 1949–1972. W orld Sci. Publ., Hackensack, NJ, 2018

work page 2018