Multifractal analysis of intermingled basins and blowout bifurcations in a parametetric family of skew product maps

Fatemeh Helen Ghane; Marc Kesseb\"ohmer

REVIEW 3 major objections 6 minor 14 references

Multifractal analysis of intermingled basins and blowout bifurcations in a parametetric family of skew product maps

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read For an open region of parameters, a two-parameter skew product family has two chaotic attractors whose basins intermingle, separated by a single invariant graph whose stability-index level sets have dimensions given by a Legendre transform.

desk verdict A solid explicit example of intermingled basins, but the multifractal spectrum is not proved as written; send to review with Section 5 flagged. read the letter →

arxiv 2502.04869 v1 pith:MIA2NGZ4 submitted 2025-02-07 nlin.CD math.DS

Fatemeh Helen Ghane , Marc Kesseböhmer This is my paper · ORCID

classification nlin.CDmath.DS MSC 37C7037C4037H1537C45

keywords invariantgraphchaoticMilnorattractorriddledbasinintermingledblowoutbifurcationstabilityindexmultifractalanalysisthermodynamicformalism

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 studies the family $F_{a,b}(x,y)=(2x\bmod 1,\,g_{a,b,x}(y))$ on the unit square, where the fiber map pushes up or down depending on which half of the square $x$ lies in. It tries to establish that for an open region $\Gamma$ in the $(a,b)$-plane the two invariant lines $y=0$ and $y=1$ each support a chaotic Milnor attractor, and that the two basins are intermingled: every neighbourhood of almost every point in one basin contains positive measure of the other basin. It further claims that a single invariant graph $\phi^*$ separates the basins, that crossing the boundary curves of $\Gamma$ produces blowout bifurcations in which one attractor becomes a chaotic saddle, and that the local stability index and the Hausdorff dimension of its level sets are given by explicit thermodynamic-pressure formulas. A sympathetic reader would care because this gives a concrete, two-parameter model in which riddled basins, intermingled basins, and their multifractal spectra are described quantitatively rather than only observed.

What carries the argument

The central object is the skew product map $F_{a,b}(x,y)=(f(x),g_{a,b,x}(y))$ with base map $f(x)=2x\bmod 1$ and fiber maps $g_a(y)=y+ay(1-y)$ for $x\in[0,1/2)$ and $g_b(y)=y-by(1-y)$ for $x\in[1/2,1]$. The two constant graphs $\phi_0(x)=0$ and $\phi_1(x)=1$ are invariant, and the proof of Theorem 1 rests on the normal Lyapunov exponents of the SRB measures on them: for $\Gamma_0$, $\Lambda_{\mathrm{SRB}}(A_0)=\tfrac12\log(1+a)+\tfrac12\log(1-b)<0$, and for $\Gamma_1$, $\Lambda_{\mathrm{SRB}}(A_1)=\tfrac12\log(1-a)+\tfrac12\log(1+b)<0$. Riddling is obtained by applying a cited criterion that combines negative SRB normal exponent with a dense set of periodic points having positive normal exponent; the separating graph $\phi^*$ is identified with the common value of the lower boundary function $\ell_{a,b}(x)=\sup\{y:\lim_n g^n_{a,b,x}(y)=0\}$ and the upper boundary function $r_{a,b}(x)=\inf\{y:\lim_n g^n_{a,b,x}(y)=1\}$, whose equality is imported from a cited proposition. The stability-index and multifractal results are carried by the pressure functions $p_{i,\psi}(t)$ and by the implicitly defined curves $T(q)$, $S(q)$, whose Legendre transforms give the Hausdorff dimensions of the level sets.

What would settle it

Compute $\ell_{a,b}(x)$ and $r_{a,b}(x)$ numerically for a dense grid of $x$ and for fine parameter grids inside $\Gamma$; if for some set of positive $\nu_{\mathrm{ac}}$-measure the two functions differ, the separating graph $\phi^*$ of Theorem 1 fails and the dimension formulas collapse. A cheaper check is to measure the stability index at points just above and below the numerically located boundary for a fixed parameter in $\Gamma$ and compare the values with the constants predicted by Theorem 4.

Watch

Extended reading notes

Core claim

For $F_{a,b}$ defined by (2.2)--(2.3), if $(a,b)$ lies in $\Gamma=\{0<a<1/2,\,0<b<1/2,\,b>a/(1+a),\,b<a/(1-a)\}$, then $F_{a,b}$ has two chaotic essential Milnor attractors $A_0\subset I\times\{0\}$ and $A_1\subset I\times\{1\}$ whose basins are locally riddled and intermingled (Theorem 1). The basins are separated by a $\nu_{\mathrm{ac}}$-almost everywhere invariant graph $\phi^*$ with $0<\phi^*<1$. Crossing $b=a/(1+a)$ or $b=a/(1-a)$ turns one of the attractors into a chaotic saddle and produces a blowout bifurcation, one supercritical and one subcritical (Theorem 2 and Corollary 2.13). Under condition (H1), the stability index $\sigma_\nu(x,y)$ is constant $\nu_\psi$-almost everywhere on each side of $\phi^*$, with values written in terms of positive zeros $t^*_0,t^*_1$ of pressure functions $p_{i,\psi}(t)=P(\psi+t\log dg_{a,b}(\phi_i))$ (Theorems 3 and 4). The level sets $A_{i,\nu}(\sigma)$ have Hausdorff dimension $T(q)+q\sigma(q)$ or $S(q)+q\sigma(q)$, where $T$ and $S$ are defined implicitly by pressure equations, so the multifractal spectrum of the stability index is the Legendre transform of a thermodynamic function (Theorems 5 and 6).

Load-bearing premise

The load-bearing premise is that for parameters in $\Gamma$ the lower and upper boundary functions $\ell_{a,b}(x)$ and $r_{a,b}(x)$ coincide almost everywhere, so that one separating graph $\phi^*$ exists, and that condition (H1) holds: a Gibbs measure $\nu_\psi$ with negative normal Lyapunov exponents on both invariant lines; the paper verifies the equality only by importing a cited proposition, and verifies (H1) explicitly only for the SRB potential.

Editorial extensions

If this is right

For every $(a,b)\in\Gamma$, almost every point of the square is eventually attracted to one of the two invariant lines $y=0$ or $y=1$, yet each basin has positive measure and is riddled with the other, so the asymptotic future of a point cannot be inferred from the futures of arbitrarily close neighbours.
The parameter plane splits into three dynamical regimes, with $b=a/(1+a)$ and $b=a/(1-a)$ as the two blowout bifurcation curves; crossing one turns the corresponding attractor into a chaotic saddle while the other attractor retains a locally riddled basin.
For $\nu_\psi$-almost every $x$, the stability index $\sigma_\nu(x,y)$ is constant on each side of the separating graph $\phi^*$, with explicit values $t^*_0\,(-\lambda_{\nu_\psi}(\phi_0))/\int\log|df|\,d\nu_\psi$ and $t^*_1\,(-\lambda_{\nu_\psi}(\phi_1))/\int\log|df|\,d\nu_\psi$.
The Hausdorff dimension of the stability-index level sets equals $T(q)+q\sigma(q)$ for the $A_0$ side and $S(q)+q\sigma(q)$ for the $A_1$ side, so the full multifractal spectrum is described by the Legendre transform of a pressure function.
In the SRB case $\psi=-\log|df|$, the stability-index theorems reduce to the previously known intermingled-basin results, confirming that the new pressure-based formulas contain the earlier theory as a special case.

Reading between the lines

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

The same two-parameter mechanism should work for any piecewise expanding Markov base map with two branches whose fiber maps have negative Schwarzian derivative, since the paper's Remark 2.1 states the arguments extend and the only quantitative input is the normal Lyapunov exponents.
If the stability index is measured numerically at points just above and below the numerically located boundary curve for a parameter inside $\Gamma$, the predicted constant values from Theorem 4 give a direct, quantitative test of the thermodynamic formalism.
The same pressure-based Legendre-transform construction could be applied to other skew products with finitely many invariant graphs to produce dimension spectra for riddled basins, although the existence of a single separating graph would need to be established separately in each case.
If condition (H1) fails for a potential other than $-\log|df|$, the dimension formulas may still hold for the SRB measure, but the claimed generality over Gibbs measures would require an additional argument.

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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

Bottom line: the paper is worth taking seriously as an explicit worked example, and the intermingled-basin/chaotic-saddle part (Theorems 1 and 2) largely holds up. The multifractal part (Theorems 5 and 6) does not: equation (5.3) is an unproved identification, and the dimension formulas are not established as written. If you cite this paper, cite it for the example and the parameter regions, not for the multifractal spectra.

What's genuinely new: the two-parameter family with exact regions Γ0, Γ1, Γ and the blowout curves b = a/(1 ± a). The proofs of Theorems 1 and 2 are direct: explicit derivative computations give the SRB normal Lyapunov exponents, and the dense-periodic-point argument supplies the local riddling. The paper is also honest about what it imports from Keller, and Remark 2.16 correctly notes that Theorems 3 and 4 reduce to [Kel17] in the SRB case. That is the right way to use prior work.

The soft spots are concentrated in Section 5. The sets A_{i,ν}(σ) are defined pointwise: σν(x,y) = σ for every y below or above φ*. Theorem 4 only determines σν for ν-a.e. x. For any σ that is not the single ergodic value, the set is a ν-null set, and no pointwise control is given. So (5.3), which identifies A_{i,ν}(σ) with a Birkhoff-average set, is an additional assertion, not a consequence of Birkhoff's theorem. Relatedly, in the proof of Theorem 6 the paper invokes [Kel17, Prop. 1.6] for νq after checking only λνq(φ1) < 0; that proposition requires both bounding graphs to have negative normal Lyapunov exponents, and λνq(φ0) < 0 is not shown. So even in the SRB case the proof does not close. The abstract's "fractal boundary curve" is also stronger than what is proved: φ* is a measurable invariant graph, with no fractal dimension or regularity established. And the subcritical/supercritical labels on the blowout bifurcations are asserted from the sign of Λ_SRB; the bifurcation type is not actually analyzed. These are fixable, but they are real gaps, not cosmetic ones.

Who gets value: anyone working on riddled or intermingled basins who wants a clean explicit family with computable parameter boundaries. The example deserves to be known. But the multifractal claims should not be relied on until the Section 5 gaps are addressed.

Recommendation: send it to peer review, with a referee specifically asked to check Section 5. A major revision is needed – either prove (5.3) with genuine pointwise control, or restrict the claims and drop the full Legendre spectrum.

Referee Report

3 major / 6 minor

Summary. The paper studies a two-parameter family of skew-product maps F_{a,b}(x,y)=(2x mod 1, g_{a,b,x}(y)) with two invariant lines. It identifies an open region Γ in the (a,b)-plane in which the two invariant chaotic attractors have intermingled basins separated by an invariant graph φ*, proves the complementary chaotic-saddle regime, and locates blowout bifurcations on the two boundary curves. The second half uses thermodynamic formalism to compute Loynes exponents and the stability index for Gibbs measures satisfying condition (H1), and claims multifractal spectra of the stability-index level sets as Legendre transforms of pressure functions.

Significance. Theorems 1 and 2 rest on explicit derivative computations and give a clean, checkable example of intermingled basins and blowout bifurcations. The paper is also honest that the existence of the separating graph is imported from Keller's work. No parameters are fitted: the stability index and the pressure functions are defined directly from the dynamics. If the multifractal dimension formulas were established, they would be a valuable concrete illustration of the Legendre-transform formalism for stability-index level sets. As it stands, however, the multifractal part contains a load-bearing gap, so the full significance claimed in the abstract is not yet realized.

major comments (3)

[Section 5, Eq. (5.3)] The displayed identification of A_{i,νψ}(σ) with a Birkhoff-average set is asserted rather than proved, and as written it is also incorrect in sign. For νψ-a.e. x and y < φ*(x), Theorem 4(2) gives σνψ(x,y) > 0, whereas the quotient Q0(x) = lim t*0 S_n log dg(φ0) / (-S_n log|df|) converges to t*0 λνψ(φ0) / ∫ log|df| dνψ < 0. Thus A0,νψ(σ) would need to be identified with {Q0 = -σ}, not {Q0 = σ}. The same sign issue occurs for A1 because (2.17) defines A1,νψ(σ) through σνψ = -σ. Moreover, Theorem 4 only computes the stability index on a νψ-full-measure set via the sequence (4.14); for σ different from the ergodic value the level set is νψ-null, and no pointwise estimates on that null set are supplied. Therefore (5.3) is an additional load-bearing assertion, and Theorems 5(2) and 6(2) do not follow from the preceding estimates.
[Section 5.1, proof of Theorem 6] In Case (1) the paper concludes from ∫ log dg(φ1) dνq < 0 that φ* is defined νq-a.e., citing [Kel17, Proposition 1.6]. In the earlier use of Proposition 1.6, namely Corollary 2.12, the condition is applied after verifying negative Lyapunov exponents for both bounding graphs φ0 and φ1 with respect to the relevant measure. For q ≠ 0 the measure νq differs from νψ, and the text does not verify λνq(φ0) < 0. Hence the use of Proposition 1.6 is missing a hypothesis, and the Moran-cover argument that relies on the νq-defined graph is unsupported. The same omission affects Theorem 5, whose proof is only said to be similar.
[Section 5, proof of Theorems 5 and 6] The theorems are stated as dimension formulas for the stability-index level sets A0,νψ(σ) and A1,νψ(σ), but the proof only attempts to analyze Birkhoff-average level sets after replacing them by (5.3). Even after correcting the sign, the paper would need to prove that the stability-index condition 'for every y < φ*(x)' is equivalent to a pointwise Birkhoff-average condition. No such equivalence is shown: Theorem 4 controls σνψ only for νψ-a.e. x and all y in the respective half-neighborhood, and the exceptional null sets are not controlled. The final dimension claims therefore need a genuinely new argument or a restricted statement, e.g. for the Birkhoff-average sets defined in (5.3).

minor comments (6)

[Section 4.2, Case (2)] The formula 'σ0ν(x,y) = t*1 · -log λ_{a,b,ν}(φ1) / ∫ log|df| dν' should read '-λ_{a,b,ν}(φ1)' instead of '-log λ_{a,b,ν}(φ1)'.
[Section 4, Proposition 4.7] Hypothesis (i) is mis-stated: the integral should be written as ∫ e^{t log dg^n_{a,b,x}(φ0)} dν, and the claimed bound cannot follow from boundedness of dg alone; the text should specify the intended inequality and its domain in t.
[Section 4, Lemma 4.5 and Corollary 4.6] There are minor typos: 'Lemm 12.1.4' appears twice, and 'P(psi)' should be 'P(ψ)'.
[Theorem 4 statement] The chain of equalities in item (1) is confusing because σ0νψ is used with different signs in the statement and in the proof; the notation should be aligned so that σνψ = σ1νψ - σ0νψ is transparent in both places.
[Section 2, Condition (H1)] Condition (H1) is verified only for the SRB potential ψ = -log|df|. The paper should state explicitly that Theorems 3-6 for general ψ are conditional on this unverified hypothesis, or provide an example of a non-SRB potential satisfying it.
[Title and abstract] The word 'parametetric' in the title is a typo, and in the abstract 'these attractors either exhibits' should be 'these attractors either exhibit'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper's derived quantities are computed from the dynamics and external results, not fitted or defined into the conclusions; the main gaps are unproved identifications, not circular reductions.

full rationale

The derivation chain is not circular in the load-bearing sense. The normal Lyapunov exponents, pressure functions p_{i,psi}(t), Loynes exponents t*_i, and stability indices are computed from the explicit skew-product dynamics (2.2)-(2.3) and thermodynamic formalism, not fitted to the quantities they are claimed to predict. The intermingled-basin and separating-graph facts are imported from Keller's independent work ([Kel17, Proposition 1.6 and 2.2]) rather than from the authors' own prior results; none of the authors' self-citations ([FKG18], [KG20], [Rab+22], [KS04], [KS07], [KS08], [JK11], [JKM21]) is invoked as the proof of a central theorem. Theorems 5 and 6 apply the standard Pesin-Weiss multifractal formalism to pressure-defined functions T and S, so the Legendre-transform dimension formulas are consequences of that formalism rather than restatements of the definitions. The skeptical concerns are genuine correctness gaps, not circularity: equation (5.3) identifies the stability-index level sets A_{i,nu_psi}(sigma) with Birkhoff-average sets only for nu_psi-a.e. points, and for non-ergodic values sigma the sets are nu_psi-null, so no pointwise estimate is supplied; Section 5.1 also verifies only one of the two normal Lyapunov exponents needed for [Kel17, Proposition 1.6]. These omissions mean Theorems 5 and 6 are not fully established, but they do not make the argument circular: the equality (5.3) is asserted, not built into the definition of the stability index, and condition (H1) is stated as a hypothesis rather than smuggled in as a conclusion. Accordingly the circularity score is low.

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

The paper introduces no new fitted constants or postulated entities. Its results rest on the standard thermodynamic formalism for expanding Markov maps, on several propositions imported from Keller [Kel17], and on the standing condition (H1), which is verified only in the SRB case. The model parameters a,b are variables, not free parameters.

assumptions (5)

standard math Existence, uniqueness, analyticity and convexity of equilibrium states and pressure for Holder continuous potentials on expanding Markov maps (Lemma 4.1).
Used throughout Sections 4 and 5 to define t*_i, T(q), S(q) and the Legendre transforms; cited from [PW97], [Rue04] and [VO16].
domain assumption The family F_{a,b} with f = 2x mod 1 and fiber maps g_a(y) = y + a y(1-y), g_b(y) = y - b y(1-y), a,b in (0,1/2), has the stated invariant sets and SRB measure nu_ac equal to Lebesgue on each branch.
Definition (2.2)-(2.3), Remark 2.3 and Section 3.1. The whole bifurcation picture is specific to this family.
ad hoc to paper Condition (H1): existence of an equilibrium state nu_psi with P(psi)=0 and lambda_{nu_psi,a,b}(phi_i) < 0 for i=0,1.
Stated in Section 2.3; used for Theorems 3-6. Only the SRB case psi = -log|df| is shown to satisfy it.
standard math [Kel17, Proposition 1.6] and [Kel17, Proposition 2.2] guarantee that if the two bounding graphs have negative normal exponents and positive intermingling, then l_{a,b} = r_{a,b} nu-a.e. and the basins are intermingled.
Invoked in Corollary 2.12, Remark 2.14, Lemma 3.5 and throughout Section 4; these results are not re-proved in the paper.
standard math Koebe-type distortion estimate (4.10) and the distortion bounds (4.1)-(4.2) hold for the family.
Assumed in Lemma 4.3, Lemma 4.10 and the proof of Theorem 3, cited to [OSY09] and [Kel17].

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Pith. "Pith review of Multifractal analysis of intermingled basins and blowout bifurcations in a parametetric family of skew product maps." pith.science (2026). https://pith.science/paper/MIA2NGZ4

@misc{pith2026250204869,
  author       = {Pith},
  title        = {Pith review of: Multifractal analysis of intermingled basins and blowout bifurcations in a parametetric family of skew product maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MIA2NGZ4}},
  note         = {Machine review of arXiv:2502.04869}
}

read the original abstract

In this paper we study a two-parameter family of planar maps characterized by two distinct invariant subspaces. The model reveals the existence of two chaotic attractors within these subspaces. We identify parameter values at which these attractors either exhibits a locally riddled basin of attraction or transitions into a chaotic saddle. In particular, we demonstrate that, for an open region in the parameter plane, their basins are intermingled. It is shown that a fractal boundary curve separates the basins of attraction of these two chaotic attractors, providing a detailed characterization of the riddled basin structure. Additionally, we show that the model undergoes a blowout bifurcation. An estimation of the stability index is examined using thermodynamic formalism. We also perform a multifractal analysis of the level sets of the stability index.

Figures

Figures reproduced from arXiv: 2502.04869 by the authors.

**Figure 1.** This plot represents the fiber maps ga (solid) and gb (dashed) for a = b = 2/5. 2.3. Invariant Graphs and Lyapunov exponents. Invariant graphs are essential objects in the study of skew product systems and are of significant interest. They often arise in systems where a base dynamics drives or interacts with a dependent (fiber) dynamics. A thorough understanding of invariant graphs in skew product systems is well-e… view at source ↗

**Figure 2.** The dashed curve represents the function a 7→ a/(1 + a) and the solid black curve represents the function a 7→ a/(1 − a). The area above the lower and below the upper curve are hatched differently and define the regions Γ0 and Γ1 in the ab-plane. The intersection of these areas are denoted by Γ. For each (a, b) ∈ Γ, both invariant sets A0 and A1 of the skew product system Fa,b are Milnor attractors with locally ridd… view at source ↗

**Figure 3.** The intermingled basins for the chaotic attractors A0 and A1 are shown for the parameters a = 0.45 and b = 0.33 and with respect to the SRB measure νac. The black points corresponds to the basin of attraction B(A0), while the white region corresponds to the basin of attraction B(A1) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The intermingled basins for the chaotic attractors A0 and A1 are shown for the parameters a = 0.45 and b = 0.33, but in this case, we consider a Gibbs measure ν that assigns weights of 1/3 and 2/3 to the intervals [0,1/2] and [1/2,1], respectively. The black points cor…

**Figure 5.** Figure 5: The separating graph ϕ ∗ for the parameters a = 0.49, b = 0.49. Lemma 4.1. Let f : I → I be a Hölder continuous piecewise expanding map, and let ψ : I → R be a Hölder continuous potential. Then there exists a unique equilibrium state νψ with respect to the potential fu…

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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Ashwin, P

[AAN98] P. Ashwin, P. J. Aston, and M. Nicol. On the unfolding of a blowout bifurcation.Physica D: Nonlinear Phenomena111.1-4 (1998), 81–95. [ABS94] P. Ashwin, J. Buescu, and I. Stewart. Bubbling of attractors and synchronisation of chaotic oscillators.Physics Letters A193.2 (1994), 126–139. [ABS96] P. Ashwin, J. Buescu, and I. Stewart. From attractor to ...

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