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arxiv: 2604.21048 · v1 · submitted 2026-04-22 · 🧮 math.DS

Emergence of Mandelbrot-like and Julia-like Structures in Parameter Slices of Rational Maps

Pedro Iv\'an Su\'arez Navarro This is my paper

Pith reviewed 2026-05-09 22:32 UTC · model grok-4.3

classification 🧮 math.DS
keywords rational mapsparameter slicesMandelbrot setJulia setconnectedness locuscritical orbitscubic polynomials
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The pith

Certain parameter slices of rational maps contain embedded Mandelbrot-like sets and Julia-like structures.

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

The paper examines one-dimensional slices within a three-parameter family of rational maps that have two free critical points. These slices are obtained by requiring the existence of periodic orbits with fixed multipliers, allowing explicit parametrization. Numerical analysis of critical orbit behavior and connectedness loci in these slices uncovers structures that closely resemble the Mandelbrot set and other features from cubic polynomial parameter spaces. This matters because it indicates that well-known fractal patterns from polynomial dynamics can arise in the more general context of rational maps through natural restrictions of the parameter space.

Core claim

By studying slices of the parameter space for rational maps where periodic orbits have prescribed multipliers, the computations show Mandelbrot-like sets in the connectedness loci and Julia-like structures from the interaction of bounded and escaping critical orbits. These regions exhibit geometric and dynamical features consistent with embedded copies of the period-one and period-two slices of cubic polynomial families.

What carries the argument

The explicit parametrizations of parameter slices obtained by fixing multipliers of periodic orbits, used to numerically approximate connectedness loci via critical orbit analysis.

If this is right

  • The slices contain subsets whose geometric features match those in classical polynomial parameter spaces.
  • Julia-like structures arise in parameter space from the differing behaviors of the two critical orbits.
  • The observed analogies apply to both period-one and period-two slices of the cubic polynomial family.
  • Classical polynomial structures emerge naturally when restricting rational maps to these particular parameter slices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the embeddings persist under refinement, similar slices could be defined in other rational families to locate higher-period Mandelbrot copies.
  • Some questions about rational-map dynamics might reduce locally to the study of their polynomial-like subregions.

Load-bearing premise

The numerical approximations of connectedness loci and critical-orbit behavior accurately capture the true dynamical structures without artifacts from finite iteration counts, discretization, or choice of escape radius.

What would settle it

A higher-resolution computation or analytic proof showing that a supposed Mandelbrot-like connected component in one of the slices is actually disconnected or lacks the expected hyperbolic components would falsify the reported analogy.

Figures

Figures reproduced from arXiv: 2604.21048 by Pedro Iv\'an Su\'arez Navarro.

Figure 1
Figure 1. Figure 1: Comparison between parameter slices in the rational family stud￾ied in this paper (top) and in the cubic polynomial family (bottom). The close visual resemblance, including Mandelbrot-like and Julia-like structures, suggests that parts of the rational parameter slices may exhibit dynamics anal￾ogous to those of cubic polynomial families. The article is organized as follows. In Section 2, we introduce the f… view at source ↗
Figure 2
Figure 2. Figure 2: Connectedness locus of S1(0) for large d, compared with the cubic slice Per1(0). For each d, this produces a finite collection of distinguished parameters, whose number grows linearly with the degree. Numerically, these points appear to organize the surrounding geometry: the visible hyperbolic components are arranged around them, and finer filamentary structures tend to accumulate near their locations. Thi… view at source ↗
Figure 3
Figure 3. Figure 3: Connectedness locus of S2(0) in the a-parameter plane for d = 3. explore the geometry of the corresponding connectedness loci and analyze how their structure varies with λ and d. Particular attention is given to the emergence of recurrent patterns and their relation to known features in parabolic slices of cubic polynomial parameter spaces. We distinguish between two cases, according to the arithmetic natu… view at source ↗
Figure 4
Figure 4. Figure 4: Connectedness locus of S2(0) for d = 3 in the parameter plane given by a−i a+i , illustrating its similarity with the corresponding slice Per2(0) of cubic polynomials. A similar organization is observed in parabolic slices of cubic polynomials of the form e 2πiθz +bz2 +z 3 , suggesting that these structures persist across degrees and may be governed by common underlying mechanisms. See also [27] for relate… view at source ↗
Figure 5
Figure 5. Figure 5: Representative Julia sets for parameters from components of types A, B, and D in the right-hand side of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Zoomed views of S2(0) from [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Connectedness loci Mλ in S1(λ) for λ = e 2πiθ, where θ ∈ Q. Accessing conjectural objects such as the Jordan curves in the bifurcation loci arising in the work of Zakeri [25] through numerical methods appears to be a challenging problem. To the best of our knowledge, no effective algorithm is currently available for visualizing such objects at high resolution in this setting, much less in more general cont… view at source ↗
Figure 8
Figure 8. Figure 8: Connectedness locus of S1(λ) in the a-parameter plane for d = 3 with θ = √ 5−1 2 . The parameter space exhibits intricate filamentary structures and fine-scale patterns, consistent with the complex behavior associated with irrational rotation numbers satisfying the Brjuno condition. Our goal is to compare the observed structures with the period-two slices Per2(λ) in the parabolic setting of cubic polynomia… view at source ↗
Figure 9
Figure 9. Figure 9: Connectedness loci Mθ for θ = 1 2 , 1 3 , 1 11 and 1 79 . The parameter space exhibits a decomposition into structured regions consistent with hyper￾bolic components, together with increasingly intricate boundaries as the de￾nominator of θ grows [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Zoomed-in views illustrating the fine-scale structure of the pa￾rameter space in [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Connectedness locus Mθ in the slice S2(θ) for d = 3 and θ = √ 5−1 2 . belong to a resonance region if at least one critical orbit converges to an attracting periodic cycle on S 1 . In this parameter space, one observes tongue-like regions corresponding to attracting peri￾odic dynamics. These tongues organize the parameter plane and reflect resonance phenomena associated with non-invertible circle covering… view at source ↗
Figure 12
Figure 12. Figure 12: Parameter space of the Blaschke slice (|a| = |b| = 1), parametrized by (e 2πiω1 , e2πiω2 ) ∈ T 2 . In the present setting, these structures arise from the interaction between the two critical orbits. This is particularly transparent in the Blaschke slice, where both critical points lie on S 1 and directly influence the formation of resonance regions and their internal bifurcation structure. An example of … view at source ↗
Figure 13
Figure 13. Figure 13: Zoom of a shrimp-like structure in (ω1, ω2)-parameter plane, il￾lustrating its internal organization and the presence of bifurcation curves de￾limiting stability regions [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Periodic island for |a| ≈ 1 and |b| ≈ 1, illustrating an isolated stability region disconnected from the main structures. 4. Concluding remarks The numerical experiments presented in this paper reveal several recurring features across the parameter slices associated with the family (2.1). In both S1(λ) and S2(λ), we observe the emergence of Mandelbrot-like structures near singular parameters, as well as r… view at source ↗
read the original abstract

We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit parametrizations, we explore these slices numerically by analyzing the behavior of the critical orbits and approximating the corresponding connectedness loci. The computations reveal rich parameter space structures closely analogous to those arising in cubic polynomial families, including Mandelbrot-like sets. In addition, we observe regions exhibiting Julia-like structures embedded in parameter space, arising from the interaction between bounded and escaping critical orbits. While the appearance of such structures is well established in polynomial dynamics, it remains comparatively less explored in the setting of rational maps. Our results provide numerical evidence that these parameter slices contain subsets closely related to the period-one and period-two slices of cubic polynomial families. More precisely, certain regions appear to exhibit geometric and dynamical features consistent with embedded copies of these classical parameter spaces. These observations highlight how classical structures from polynomial dynamics can emerge naturally within parameter slices of rational maps.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper studies one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, constructed by imposing periodic orbits with prescribed multipliers. Using explicit parametrizations, the authors numerically analyze critical-orbit behavior and approximate the connectedness loci in these slices, reporting the emergence of Mandelbrot-like sets and Julia-like structures analogous to those in cubic polynomial families, including embedded copies of the period-one and period-two slices.

Significance. If the numerical observations prove robust, the work would be significant in showing that classical polynomial-dynamics structures can arise naturally in rational-map parameter spaces, a setting where such phenomena remain comparatively unexplored. The explicit parametrizations and focus on critical-orbit interactions provide a concrete computational bridge between the two classes of maps and suggest testable predictions for further study.

major comments (1)
  1. [Abstract] Abstract: the central claim that the slices contain 'embedded copies' of cubic period-one and period-two slices rests entirely on numerical approximations of connectedness loci and critical-orbit behavior, yet the abstract (and the provided description of the computations) supplies no iteration counts, escape-radius values, grid resolution, or convergence tests. This omission is load-bearing because, as the stress-test note observes, small changes in these parameters can alter apparent connectivity near boundaries and introduce discretization artifacts.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We address the major comment below and will revise the manuscript to incorporate additional details on our numerical methods, thereby improving transparency and reproducibility.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the slices contain 'embedded copies' of cubic period-one and period-two slices rests entirely on numerical approximations of connectedness loci and critical-orbit behavior, yet the abstract (and the provided description of the computations) supplies no iteration counts, escape-radius values, grid resolution, or convergence tests. This omission is load-bearing because, as the stress-test note observes, small changes in these parameters can alter apparent connectivity near boundaries and introduce discretization artifacts.

    Authors: We agree that the abstract and computational description would benefit from explicit numerical parameters to allow readers to evaluate the robustness of the observed structures. Our results are presented as numerical evidence of analogous behavior rather than rigorous proofs, and the absence of these specifics was an oversight in the presentation. In the revised manuscript we will add the following details: critical orbits are tracked for 500--2000 iterations (depending on the slice and escape behavior), with an escape radius of 10^4; parameter slices are discretized on grids of at least 1000 x 1000 points; and we include a brief description of convergence checks and boundary stress tests (varying iteration count and escape radius by factors of 2--5) to confirm that apparent connectivity is not an artifact. These additions will be placed in both the abstract and the methods section without changing the stated conclusions. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from complex dynamics for interpreting critical-orbit behavior as determining connectedness loci, plus the domain assumption that the imposed periodic-orbit conditions correctly define the desired one-dimensional slices. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Behavior of critical orbits determines the connectedness locus in parameter space
    The numerical exploration approximates connectedness loci by tracking bounded versus escaping critical orbits, invoking this established principle of complex dynamics.
  • domain assumption Imposing periodic orbits with prescribed multipliers yields valid one-dimensional slices of the three-parameter family
    The family is constructed by this imposition, which is taken as given to reduce the parameter space.

pith-pipeline@v0.9.0 · 5475 in / 1390 out tokens · 98120 ms · 2026-05-09T22:32:43.742167+00:00 · methodology

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Reference graph

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