Hausdorff Dimension of Sets of Generic Points for Non-statistical Dynamical Systems

Douglas Coates; Katrin Gelfert

REVIEW 2 major objections 5 minor 25 references

Hausdorff Dimension of Sets of Generic Points for Non-statistical Dynamical Systems

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For one-dimensional maps with several neutral fixed points, every basin of attraction of a measure in the simplex of fixed-point measures, and every set whose empirical measures accumulate on a prescribed closed connected set, has Hausdorff

desk verdict Full Hausdorff dimension for basins of non-ergodic measures in non-statistical interval maps; a solid, modular proof whose only real gap is a terse appeal to Sera's functional limit theorem. read the letter →

arxiv 2509.00241 v2 pith:XYUZSWON submitted 2025-08-29 math.DS

Douglas Coates , Katrin Gelfert This is my paper

classification math.DS MSC 37E0537C45

keywords Hausdorffdimensiongenericpointsnon-statisticaldynamicsneutralfixedintermittentmapsempiricalmeasuresinducedmaprepellers

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

This paper proves that for a class of one-dimensional intermittent maps with several neutral fixed points, the sets of points whose empirical measures converge to a given measure—or accumulate exactly on a prescribed closed connected set of measures—have Hausdorff dimension 1. These maps are 'non-statistical': typical orbits do not have converging empirical distributions, and no physical measure exists. Yet the basins of attraction of every measure supported on the fixed points are dimensionally maximal, even though they are Lebesgue-null and have zero topological entropy. The proof reduces the problem to return-time statistics of an expanding induced map and constructs fractal sets of full dimension inside each basin.

What carries the argument

The coding equivalence e_n(x)→ν_p iff τ_k/τ_k(x)→p transfers the problem to Birkhoff sums of the return-time vector τ=(τ^{(1)},...,τ^{(d)}) under the uniformly expanding Gibbs-Markov induced map F. The central construction builds, for each target point p and tolerance ε, a finite collection of cylinders A⊂Q^n whose return-time ratios lie in B_ε(p) and whose maximal invariant set Λ(A) (an expanding repeller for F^n) has a geometric measure with cylinder ratios close to 1; these repellers have dimension 1−O(1/n). Concatenating them along carefully chosen time scales yields a fractal Γ with pointwise dimension 1 and V(x)=πC.

What would settle it

Take an intermittent map with two neutral fixed points with different power-law exponents α_1 ≠ α_2 (satisfying Assumptions 1 and 2 but not Assumption 3(3)). If the set of points whose empirical measures accumulate on the full 2-simplex has Hausdorff dimension less than 1, or if some basin G(ν_p) has dimension less than 1, then the equal-tail assumption is necessary and the theorem's conclusion fails beyond it.

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Extended reading notes

Core claim

Theorem A: if the map satisfies Assumptions 1–3 (Markov partition, Gibbs-Markov first return map to a set Y, and identical power-law tails for the return times to each of the d neutral fixed points), then for every closed connected set C in the simplex S of convex combinations of fixed-point Dirac measures, dim_H {x : V(x)=πC} = 1, where V(x) is the set of weak-* limit points of empirical measures. In particular, each measure ν_p has a basin G(ν_p) of full Hausdorff dimension. The discovery is that non-statistical behavior does not destroy dimensional size: the generic-point sets, though negligible in Lebesgue measure and entropy, are as large as possible in Hausdorff dimension.

Load-bearing premise

All neutral fixed points must have return-time tails that decay as n^{-α} with the same exponent α; if the exponents differ, the distributional limit used in the proof would not have full support on the simplex, and the full-dimension conclusion could collapse.

Editorial extensions

If this is right

For every p in the simplex, the basin G(ν_p) has full Hausdorff dimension 1, so each of these non-ergodic measures is 'generic' for a dimensionally large set.
The conclusion holds for any closed connected set C: the set of points whose empirical measures accumulate exactly on πC has dimension 1, so the full multifractal spectrum for accumulation sets is dimensionally trivial.
Zero Lebesgue measure and zero topological entropy do not constrain Hausdorff dimension in this non-statistical setting; the generic-point sets are as large as possible.
The method supplies a geometric measure (full-dimension measure) supported on each such generic-point set, not just a dimension estimate.

Reading between the lines

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

If Assumption 3(3) were relaxed to allow different exponents α_j across fixed points, Sera's functional limit theorem would yield a limit law supported on a lower-dimensional face of the simplex; the corresponding generic-point sets would likely have Hausdorff dimension strictly below 1, making the equal-exponent hypothesis load-bearing rather than cosmetic.
The same repeller-assembly technique might compute the Hausdorff dimension of sets with prescribed accumulation sets for other non-hyperbolic systems with induced Gibbs-Markov maps, e.g., Lorenz-like maps or piecewise expanding maps with intermittent branches.
A testable extension: for maps with countably many neutral fixed points with varying exponents, the dimension of {x:V(x)=πC} may depend on the subset of fixed points that C intersects, producing a nontrivial dimension spectrum.

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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

Short version: this paper proves that for a class of one-dimensional maps with several equally sticky neutral fixed points (the non-statistical maps studied earlier by Coates–Melbourne–Talebi), the basins of the non-ergodic invariant measures at the fixed points, and more generally the sets of points whose empirical measures accumulate on any closed connected subset of the simplex, have full Hausdorff dimension 1. That is genuinely new and it is the natural completion of earlier results: these sets are null for Lebesgue and have zero topological entropy, but they are dimension-theoretically maximal.

The proof is careful and modular. The coding step (Theorem 3.1) reduces convergence of empirical measures to convergence of return-time ratios under the induced Gibbs–Markov map. The construction of repellers Λ(A(n)) in Section 5 is self-contained, with no fitted parameters; the dimension estimates come from standard conformal repeller theory. The concatenation into Γ in Sections 6–7 is explicit and the measure constructed on Γ genuinely has pointwise dimension 1.

The soft spot is Lemma 5.3, which imports Sera's functional limit theorem for occupation times and checks the hypotheses in a single sentence. The claim needs a limit law equivalent to Lebesgue on the simplex; if Sera's assumptions fail, the positive-measure lower bound collapses. The hooks cited — regular variation of the return-time tails from Assumption 3, continued-fraction mixing for the topologically mixing Gibbs–Markov induced map — are natural and likely sufficient, but "likely" is not a proof. A referee should ask for a line-by-line verification. There are also a few typos (p_1+p_2+p_2=1 in the introduction, [0,1] for the simplex in Lemma 5.3), which are cosmetic.

The equal-α condition in Assumption 3(3) is load-bearing and is explicitly assumed; it is satisfied by the examples, so it is a scope condition rather than a flaw. I do not see circularity: the cited prior results establish non-statistical behavior and mixing, not the dimension statement.

This paper is for people working in multifractal analysis and dimension theory of non-uniformly hyperbolic systems. It deserves a serious referee; my own verdict is a confident accept after the Sera verification is checked.

Referee Report

2 major / 5 minor

Summary. The paper studies one-dimensional Markov maps with d≥2 neutral fixed points and no absolutely continuous invariant probability measure (only σ-finite). Previous work [CMT24] established that for Lebesgue-a.e. x, the empirical measures e_n(x) accumulate on the full simplex πS of invariant measures supported at the neutral fixed points. The present paper proves Theorem A: under Assumptions 1–3 (Markov structure, a Gibbs–Markov induced map on a return set Y, and equal-α regularly varying return-time tails at the d fixed points), for every closed connected C⊂S the set {x: V(x)=πC} has Hausdorff dimension 1. In particular each basin G(ν_p) has full Hausdorff dimension. The proof encodes generic behaviour through return-time ratios under the induced map, constructs finite cylinder collections A(n) whose maximal invariant sets have return ratios close to a prescribed p and dimension close to 1, and then stacks these sets along a sparse sequence of times to form a Cantor–like set Γ with a measure of pointwise dimension 1. A final argument handles arbitrary connected C by a path-like sequence p_i.

Significance. If correct, the theorem provides a sharp dimensional dichotomy: for non-statistical interval maps the basins of attraction and, more generally, sets with prescribed accumulation set for the empirical measures have full Hausdorff dimension although they are Lebesgue-null and have zero topological entropy. This complements the Pfister–Sullivan entropy bound (1.9) and is a genuine multifractal result for level-2 non-generic points in a non-hyperbolic setting. The proof is modular and uses no free parameters; the main result is explicitly conditional on checkable Assumptions 1–3, and Section 8 gives a class of examples. The main tools (repeller approximation, virtual dimension, geometric measures) are standard but are combined in a novel way.

major comments (2)

[§5.1, Lemma 5.3] Lemma 5.3 is the only place an external functional limit theorem ([Ser20]) is imported, and it is load-bearing for Proposition 5.2. The verification of Sera's hypotheses is compressed into a single sentence. Please (i) verify [Ser20, Assumptions 2.1–2.3] step by step, in particular the continued-fraction mixing condition, which is not an immediate consequence of topological mixing of F; (ii) state precisely which convergence is used—the text says convergence of e_n(x) (time n under f), but the desired lower bound concerns the return-time ratio τ_n/τ_n after n returns, and the random time change τ_n is not automatic; if [Ser20] directly applies to the ratios, say so; (iii) justify that the limiting law Z∗P is equivalent to Lebesgue measure on the simplex (or [0,1]^d), not merely a non-atomic law. Without a positive uniform lower bound, the rest of the construction collapses.
[§5.2–5.3, Lemmas 5.6–5.7] Lemmas 4.3 and 4.4 require that F^n restricted to Λ(A) be topologically mixing, but Proposition 5.2 never establishes this for the constructed A(n). The construction appears to make the transition graph on the sets {Y_1,...,Y_L} complete, so the property is likely true, but it must be proved explicitly or cited. Without mixing, the virtual-dimension error bound and the existence of the geometric measure m are not justified.

minor comments (5)

[Introduction, page 4] In the definition of S, 'p_1+p_2+p_2=1' should be 'p_1+p_2+p_3=1'.
[§2.1, Assumption 3(3)] The asymptotic is written 'μ_Y(τ^{(j)}>n)∼γ_j nα'; the exponent should be n^{-α}.
[§5.1, Lemma 5.3] The random variable Z is said to have distribution on '[0,1]' but the vector (e_n(X_1),...,e_n(X_d)) takes values in the simplex; the target space should be corrected.
[§5.1, equation (5.4)] The bound '≤2M/n' is slightly optimistic: the denominator contains τ_{n-k0}(b a_{sj}), so the correct estimate is 2M/(n-k0) (or similar). The argument still works for large n, but the displayed inequality is not literally correct.
[§7.1, first paragraph] The existence of a sequence (p_i)⊂C with limit points equal to C and |p_i−p_{i+1}|→0 is asserted for every closed connected C. This is true for continua, but a short proof or reference would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is derived from explicit assumptions via a self-contained construction; self-citations are contextual only.

full rationale

Theorem A is conditional on Assumptions 1–3. The proof (Sections 3–7) is a constructive existence argument: it builds a set Γ by concatenating cylinders from repellers A_i, proves Γ⊂{x:V(x)=πC} via the coding Theorem 3.1 (proved in the paper from the Portmanteau theorem and elementary return-time inequalities), and proves dim_H Γ=1 via the mass distribution principle applied to a measure constructed from geometric measures. The lower bound for the dimension of the intermediate repellers (Lemmas 5.3, 5.6) uses Sera's functional limit theorem [Ser20] and standard conformal-repeller estimates [PT93], [PU10]; these are external results that do not assume the dimension conclusion. The growth sequence (k_i) is chosen recursively to satisfy the inequalities (6.9)–(6.15); it is not fitted to the target dimension. The paper's own prior work [CMT24], [CM25] is used only to describe the motivating non-statistical phenomenon and to verify that the examples in Section 8 satisfy Assumptions 1–3; neither use is load-bearing for the proof of Theorem A, which goes through for any system satisfying the assumptions. The equal-exponent condition in Assumption 3(3) is a stated scope condition, not a hidden restatement of the conclusion. No equation in the paper defines the target dimension in terms of the input, and no fitted parameter is renamed as a prediction. Hence no circular step is present.

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

No fitted parameters, no new entities. The constants (C, C', M_i, etc.) are constructive bounds, not free parameters. The result rests on the stated assumptions and on standard or external theorems.

assumptions (4)

domain assumption Assumptions 1-3: Markov structure, Gibbs-Markov first return map, and homogeneous polynomial return time tails μ_Y(τ^{(j)}>n)∼γ_j n^{-α} with common α∈(0,1).
Defines the class of maps; the common α is load-bearing for the simplex structure of limit points (Section 2.1).
domain assumption Functional limit theorem for occupation times of intermittent maps (Sera 2020, Corollary 4.2 and Theorem 3.3).
Used in Lemma 5.3 to get a uniform Lebesgue lower bound for the set of points with prescribed return-time proportions.
standard math Existence and properties of geometric measures for C^2 expanding repellers (Przytycki-Urbański, Theorems 8.1.6 and 9.1.6).
Provides the measure m satisfying (4.5) used in Section 5.3.
standard math The set of limit points of a sequence with d(e_{n+1},e_n)→0 is closed and connected (standard topological lemma).
Justifies the restriction to closed connected C in Theorem A (Introduction, p.2).

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Pith. "Pith review of Hausdorff Dimension of Sets of Generic Points for Non-statistical Dynamical Systems." pith.science (2026). https://pith.science/paper/XYUZSWON

@misc{pith2026250900241,
  author       = {Pith},
  title        = {Pith review of: Hausdorff Dimension of Sets of Generic Points for Non-statistical Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XYUZSWON}},
  note         = {Machine review of arXiv:2509.00241}
}

read the original abstract

We consider one dimensional maps with several neutral fixed points that do not admit any physical measures. We show that there is simplex of measures so that every measure in this simplex has a basin which has full Hausdorff dimension.

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Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

An introduction to infinite ergodic theory , volume 50 of Math

Jon Aaronson. An introduction to infinite ergodic theory , volume 50 of Math. Surv. Monogr. Providence, RI: American Mathematical Society, 1997

work page 1997

[2] [2]

Historic behaviour vs

Martin Andersson and Pierre-Antoine Guih\' e neuf. Historic behaviour vs. physical measures for irrational flows with multiple stopping points. Adv. Math. , 409(part A):Paper No. 108626, 71, 2022

work page 2022

[3] [3]

Occupation times of sets of infinite measure for ergodic transformations

Jon Aaronson, Maximilian Thaler, and Roland Zweim\" u ller. Occupation times of sets of infinite measure for ergodic transformations. Ergodic Theory Dynam. Systems , 25(4):959--976, 2005

work page 2005

[4] [4]

Barrientos and Raul R

Pablo G. Barrientos and Raul R. Chavez. Historical behavior of skew products and arcsine laws, 2024

work page 2024

[5] [5]

On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents

Luis Barreira, Yakov Pesin, and J \"o rg Schmeling. On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. Chaos , 7(1):27--38, 1997

work page 1997

[6] [6]

Sets of ``non-typical'' points have full topological entropy and full Hausdorff dimension

Luis Barreira and J \"o rg Schmeling. Sets of ``non-typical'' points have full topological entropy and full Hausdorff dimension. Isr. J. Math. , 116:29--70, 2000

work page 2000

[7] [7]

Topological entropy for noncompact sets

Rufus Bowen. Topological entropy for noncompact sets. Trans. Am. Math. Soc. , 184:125--136, 1974

work page 1974

[8] [8]

Persistent non-statistical dynamics in one-dimensional maps

Douglas Coates and Stefano Luzzatto. Persistent non-statistical dynamics in one-dimensional maps. Comm. Math. Phys. , 405(4):Paper No. 102, 34, 2024

work page 2024