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A note on the relation between the metric entropy and the generalized fractal dimensions of invariant measures

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For uniformly expanding maps, the full spectrum of generalized fractal dimensions of an invariant measure is pinned between entropy divided by the two expansion rates.

desk verdict The dimension-entropy inequalities are mostly sound and worth knowing, but the Section 5 settlement of Sigmund's conjecture fails on a false equality. read the letter →

arxiv 1908.00998 v2 pith:YMD4PE2J submitted 2019-08-02 math.DS

Alexander Condori , Silas L. Carvalho This is my paper · ORCID
classification math.DS MSC 37C4537B4037C4037D20
keywords expansivehomeomorphismsHausdorffdimensionpackinginvariantmeasuresgeneralizedfractaldimensionsdynamicalsystemswithspecificationmetricentropyBrin-Katoktheorem
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's central aim is a quantitative bridge between the dynamical quantity metric entropy and the geometric quantities called $q$-generalized fractal dimensions of invariant measures. It proves that when a continuous map on a compact metric space expands small distances by a factor between $\lambda$ and $\Lambda$, and the measure satisfies Brin-Katok's theorem pointwise, then every generalized dimension $D^-_\mu(q)$ and $D^+_\mu(s)$ lies between $h_\mu(f)/\log \Lambda$ and $h_\mu(f)/\log \lambda$ for $q>1$ and $s<1$. This makes metric entropy the controlling quantity for the dimension spectrum, and it gives an alternative route to Young's formula for the Bowen-Margulis measure of a $C^{1+\alpha}$ Axiom A surface diffeomorphism. The paper also derives generic zero-dimension and zero-entropy statements for expansive homeomorphisms and for Lipschitz systems with specification, including a proposed settlement of Sigmund's conjecture.

What carries the argument

The load-bearing object is the pair of uniform local dimensions $\underline d_\mu(x)$ and $\overline d_\mu(x)$, the liminf and limsup of $\log \mu(B(y,\varepsilon))/\log \varepsilon$ as $\varepsilon\to 0$ with $y$ near $x$. The argument combines two inclusions for Bowen balls under the expansion hypothesis: $B(x,\varepsilon\Lambda^{-n})\subset B(x,n,\varepsilon)$ and $B(x,n,\varepsilon)\subset B(x,\varepsilon\lambda^{-n})$, which transfer exponential rates of orbit separation into rates for $\mu(B(x,\varepsilon))$. These bounds give $\underline d_\mu(x)\ge h_\mu(f)/\log \Lambda$ and $\overline d_\mu(x)\le h_\mu(f)/\log \lambda$, and Theorem 1.2 then converts them into uniform bounds on $D^-_\mu(q)$ and $D^+_\mu(s)$ across the whole $q$-range.

What would settle it

Compute the energy integral $I_\mu(q,\varepsilon)$ for an ergodic invariant measure of a linear expanding map of a torus that satisfies the uniform expansion bounds and Brin-Katok pointwise. If the empirical lower dimension $D^-_\mu(q)$ falls below $h_\mu(f)/\log \Lambda$ or the upper dimension $D^+_\mu(s)$ exceeds $h_\mu(f)/\log \lambda$, Theorem 1.3 is false.

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Extended reading notes

Core claim

The central discovery is the chain of inequalities in Theorem 1.3: under the stated hypotheses, for every $q>1$ and $s<1$, one has $h_\mu(f)/\log \Lambda \le D^-_\mu(q) \le D^-_\mu(1) \le D^+_\mu(1) \le D^+_\mu(s) \le h_\mu(f)/\log \lambda$. The proof shows that the lower and upper local dimensions of $\mu$ are pointwise bounded by $h_\mu(f)/\log \Lambda$ and $h_\mu(f)/\log \lambda$, and then invokes a general covering argument that upgrades such uniform local-dimension bounds into bounds on every generalized dimension. Thus the entire $q$-spectrum of an invariant measure is controlled by metric entropy and the expansion constants alone, without additional multifractal data.

Load-bearing premise

The Section 5 genericity results stand on two premises: that periodic measures exhaust the space of invariant measures, and that zero-dimension genericity theorems proved for full shifts carry over unchanged to general Lipschitz systems; if either premise fails, the zero-entropy genericity claims collapse.

Editorial extensions

If this is right

  • For any $f$-homogeneous measure on a uniformly expanding system, all generalized dimensions for $q>1$ and $s<1$ are bracketed by $h_\mu(f)/\log \Lambda$ and $h_\mu(f)/\log \lambda$.
  • The Bowen-Margulis measure of a $C^{1+\alpha}$ Axiom A surface diffeomorphism satisfies $D^\pm_\mu(q)=h_\mu(T)(1/\lambda_1-1/\lambda_2)$ for every real $q$, recovering Young's formula by a new route.
  • An expansive homeomorphism with a hyperbolic metric obeys $D^+_\mu(q)\le h_\mu(f)\log k$ for $q\ge 1$ and $D^+_\mu(q)\le 2h(f)/\log k$ for $q\in[0,1)$, so dimension spectra of invariant measures are bounded by entropy and the hyperbolicity constant.
  • For $C^1$ Axiom A systems, a residual set of invariant measures has $D^+_\mu(q)=0$ for $q\ge1$, meaning typical orbits are extremely tight on small scales.
  • If an expansive homeomorphism carries a measure with $D^+_\mu(q)>0$, then the topological entropy is positive.

Reading between the lines

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

  • Beyond the paper, the bracketing may extend to measures satisfying Brin-Katok only almost everywhere, which would transfer the entropy control of dimension spectra from uniformly hyperbolic to many non-uniformly hyperbolic systems.
  • The theorem also suggests a variational formula in which the lower bound $h_\mu(f)/\log \Lambda$ plays the role of a dimension lower bound, connecting the result to Kaplan-Yorke-type estimates for non-conformal attractors.
  • A testable extension is to check numerically on piecewise expanding maps whether empirical correlation sums obey the same bracketing when the uniform expansion constants are replaced by local time-averaged expansion rates.
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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

3 major / 4 minor

Summary. The paper studies upper and lower q-generalized fractal dimensions of invariant measures for continuous maps on compact metric spaces. Its main results are: (i) Theorem 1.2, which bounds generalized dimensions by uniform lower and upper local dimension constants; (ii) Theorem 1.3, which gives a quantitative sandwich relating metric entropy and generalized dimensions under uniform expansion and contraction constants, assuming Brin–Katok's theorem holds punctually; (iii) an alternative proof of Young's formula for the generalized dimensions of the Bowen–Margulis measure of a C^{1+α} Axiom A surface diffeomorphism; (iv) estimates for generalized dimensions of invariant measures of expansive homeomorphisms under a hyperbolic metric; and (v) genericity results for zero-entropy invariant measures, claimed to settle a conjecture of Sigmund for Lipschitz maps with the specification property.

Significance. If correct, Theorems 1.2 and 1.3 would provide a clean and useful bridge between metric entropy and generalized fractal dimensions, and the alternative proof of Young's theorem would be a valuable exposition. The paper is clearly written and the dimension-entropy inequalities in Sections 2 and 4 are plausible and partly standard. However, the Section 5 claim to settle Sigmund's conjecture rests on a false implication about the specification property, and the proofs of Theorems 5.1 and 5.2 also depend on unverified extensions of results from the authors' preprints. Because this advertised contribution is invalid as stated, the paper in its current form cannot be recommended for publication.

major comments (3)
  1. [Section 5, Definition of Mco(f) and Theorems 5.1, 5.2] The assertion that the specification property implies Mco(f) = M(f) is false. Under the paper's own definition, Mco(f) is the set of periodic orbit measures, i.e., measures equidistributed on a single periodic orbit. Specification guarantees only that such measures are dense in M(f). Since M(f) is convex, any convex combination of two distinct periodic orbit measures is invariant but is not itself a periodic orbit measure; hence Mco(f) = M(f) can hold only when M(f) consists of a single periodic orbit measure. Thus the hypothesis of Theorems 5.1 and 5.2 fails for the intended examples such as the full shift or Axiom A systems, and the claimed settlement of Sigmund's conjecture collapses.
  2. [Section 5, proofs of Theorems 5.1 and 5.2] The proofs rely on Propositions 2.2 and 2.5 of the authors' preprint [6] and Theorem 1.2 of preprint [5], which are neither proved nor independently verified in this manuscript. The text asserts that these results 'can be extended' from the full-shift to the general setting, but no argument or precise statement is supplied. Since Lemma 2.1 only converts zero dimension into zero entropy, the generic zero-entropy conclusion depends entirely on these unproved transfer statements.
  3. [Section 2, proof of Theorem 1.2] The covering argument contains a gap. After choosing the finite subcover {B(x_i, epsilon(x_i))}, the proof constructs balls B(y_j, epsilon(k)) with y_j belonging to some B(x_l, epsilon(x_l)), and then applies inequality (9) to B(y, epsilon) for epsilon <= epsilon(k). However, (9) requires y to lie in B(x_l, epsilon), not merely in the larger ball B(x_l, epsilon(x_l)); the small ball B(y_j, epsilon(k)) need not be contained in any B(x_l, epsilon(x_l)). This gap is likely repairable by a standard Lebesgue-number or Vitali covering argument, but as written the proof of Theorem 1.2 is incomplete, and Theorem 1.3 depends on it.
minor comments (4)
  1. [Section 4, proof of Theorem 1.5] The line 'Thus, for q = 0' in the passage leading to equation (20) is confusing: the statement of Theorem 1.5 concerns q in [0,1), and the proof appears to specialize to q=0 after using monotonicity. Please clarify the logical order and explicitly state where the monotonicity of D^+_mu(q) is being invoked.
  2. [Section 1, Theorem 1.3 and Remark 1.1] The hypothesis 'Brin-Katok's Theorem is satisfied punctually' should be stated explicitly as a definition (e.g., the lower and upper local entropies coincide for every point and equal h_mu(f)). As written, the reader must infer the meaning from the proofs.
  3. [Section 5, Corollary 5.1] Corollary 5.1 is stated for the full shift over an uncountable alphabet; the proof refers to the same unverified extension of [6], so the corollary is not established independently of the preprint results.
  4. [Throughout] There are several minor typographical issues, including 'Hentchel' for 'Hentschel' in Definition 1.2 and the inconsistent use of 'closet' for 'closed' in Section 4. These should be corrected in a revision.

Circularity Check

3 steps flagged · score 7.0 of 10

The dimension-entropy bounds are self-contained, but Section 5's advertised settlement of Sigmund's conjecture rests on asserted extensions of the authors' own preprints and on a false specification-to-periodic-measures premise.

  1. self citation load bearing [Section 5, proof of Theorem 5.1]
    "Thus, one gets from Propositions 2.2 and 2.5 in [6] that {µ ∈ M e(f ) | dim+ H(µ ) = 0 } is a generic subset of M(f ) (although Proposition 2.2 in [6] was proven for the full-shift system presented in Subsection 1.2, the result can be extended to the dynamical system ( X, f ) considered here)."

    The key genericity statement about zero upper packing dimension is imported from the authors' own preprint [6] and its extension to the present class of Lipschitz invertible systems is merely asserted. The proof then reads off zero metric entropy from that unverified input via Lemma 2.1(i). Thus the advertised zero-entropy genericity conclusion is not independently derived; it is carried by a self-citation whose transferability is the actual load-bearing assumption.

  2. self citation load bearing [Section 5, proof of Theorem 5.2]
    "Theorem 1.2 in [5] states that, for each q ∈ (0, 1), {µ ∈ M(f ) | D− µ(q) = 0 } is a residual subset of M(f ). The result is now a consequence of Proposition 1.1 and Lemma 2.1(i)."

    Theorem 1.2 of [5] is the authors' own preprint result for full-shift systems, but it is applied here to arbitrary Lipschitz systems satisfying Mco(f)=M(f) with no proof that the full-shift genericity transfers. The proof of Theorem 5.2 therefore reduces the claimed settlement of Sigmund's conjecture to a self-cited statement whose scope is not established in the paper.

1 more flagged steps
  1. other [Section 5, after Theorem 5.2]
    "Theorem 5.2 partially settles a conjecture posed by Sigmund in [25], which states that if a topological dynamical system ( X, f ) satisfies the specification property (and consequently, Mco(f ) = M(f ); see [25]), then {µ ∈ M(f ) | hµ (f ) = 0 } is a residual subset of M(f )."

    The paper's application to specification systems depends on the assertion that specification implies Mco(f)=M(f). With Mco(f) defined as the set of periodic-orbit measures, specification only gives density of such measures in M(f), not equality; for example, the two-sided full shift on two symbols has many non-periodic invariant measures. Hence the premise of Theorems 5.1 and 5.2 is not a consequence of specification, and the claimed settlement is conditional on a condition that the intended systems do not satisfy.

full rationale

The core dimension-entropy results are not circular. Theorem 1.2 is an elementary covering argument using only the assumed uniform bounds on local dimensions and the external Proposition 1.1. Theorem 1.3 follows from Lemma 2.1, which derives pointwise dimension bounds from the assumed expansion constants and the punctual Brin-Katok identity; no fitted parameter is renamed as a prediction. The proofs of Corollary 1.2 and Theorems 1.5 and 1.6 follow standard external arguments (Young, Fathi, Brin-Katok) and are self-contained against the cited literature. The circularity is concentrated in Section 5. There, the zero-entropy genericity theorems are supported by the authors' own preprints [5] and [6], whose results are asserted to extend to the general setting without proof. Moreover, the bridge used to connect these theorems to Sigmund's specification conjecture, namely Mco(f)=M(f), is false as stated: specification yields density of periodic measures, not equality, so the intended specification systems do not satisfy the hypothesis of Theorems 5.1 and 5.2. The advertised settlement of Sigmund's conjecture therefore reduces to an unverified self-citation chain and a false premise, while the rest of the paper remains independent. Score 7 reflects that the central dimensional inequalities are sound and self-contained, but a major advertised conclusion is not.

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

The central dimension theorems rely on standard background (Brin-Katok, Fathi hyperbolic metric, Pesin's theorem). The Section 5 genericity claims additionally require unproved extensions of the authors' own preprints and a false implication about specification, which are the main weaknesses.

assumptions (5)
  • domain assumption Brin-Katok's theorem holds punctually at every x in X for the measures considered.
    Used in Theorem 1.3 to replace almost-everywhere local entropy by the global metric entropy h_mu(f) at all points, which is needed for the inequalities for all q and s.
  • standard math The hyperbolic metric d and constant k > 1 from Fathi's Theorem 1.4 exist for every expansive homeomorphism.
    Invoked in Theorems 1.5 and 1.6 to relate Bowen balls at radius r and k^{-n} r.
  • ad hoc to paper Propositions 2.2 and 2.5 in the authors' preprint [6] can be extended from the full-shift to any system with M_co(f) = M(f).
    Asserted in the proof of Theorem 5.1 without proof; load-bearing for the zero-entropy genericity result.
  • ad hoc to paper Theorem 1.2 in the authors' preprint [5] holds and applies to the systems in Theorem 5.2.
    Invoked without proof in the proof of Theorem 5.2.
  • ad hoc to paper The specification property implies M_co(f) = M(f), where M_co(f) is the set of periodic measures defined in Section 5.
    Used to claim that Theorem 5.2 partially settles Sigmund's conjecture; the implication is false as stated, since specification gives density, not equality.

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Pith. "Pith review of A note on the relation between the metric entropy and the generalized fractal dimensions of invariant measures." pith.science (2026). https://pith.science/paper/YMD4PE2J

@misc{pith2026190800998,
  author       = {Pith},
  title        = {Pith review of: A note on the relation between the metric entropy and the generalized fractal dimensions of invariant measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMD4PE2J}},
  note         = {Machine review of arXiv:1908.00998}
}
abstract

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower $q$-generalized fractal dimensions $D^{\pm}_{\mu}(q)$, $q\in\mathbb{R}$, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem~\cite{Young} for the generalized fractal dimensions of the Bowen-Margulis measure associated with a $C^{1+\alpha}$-Axiom A system over a two-dimensional compact Riemannian manifold $M$. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures such that $D_\mu^+(q)=0$ ($q\ge 1$), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\in [0,1)$, $D^{+}_{\mu}(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in~\cite{Sigmund1974} for Lipschitz transformations which satisfy the specification property.

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