Parabolic Implosion in Complex Dynamics

• Parabolic implosion is the bifurcation phenomenon where a perturbed parabolic fixed point converges, via Lavaurs scaling, to a nontrivial transition map.
• It involves critical perturbation scales that aggregate O(n) small errors, leading to explicit Lavaurs maps and enriched dynamical behavior in diverse settings.
• The theory extends from one-dimensional additive cases to non-autonomous, Möbius model, and higher-dimensional scenarios, providing practical insights into complex dynamics.

Parabolic implosion is the bifurcation phenomenon that occurs when a parabolic fixed point is perturbed so that long iterates no longer approximate the unperturbed dynamics, but instead converge to a nontrivial transition map, classically a Lavaurs map. In one complex variable, the standard local model is a germ tangent to the identity, such as $f(z)=z+z^2+O(z^3)$, perturbed by $f_\varepsilon(z)=f(z)+\varepsilon^2$; under the Lavaurs scaling, many iterates accumulate into a limiting map between attracting and repelling Fatou coordinates (Astorg et al., 29 Mar 2026). Recent work has extended this picture in three directions: to non-autonomous additive perturbations of arbitrary parabolic germs (Astorg et al., 29 Mar 2026), to an explicit Möbius-based non-autonomous model governed by orthogonal-polynomial recurrences (Huneycutt et al., 22 Jan 2026), and to genuinely higher-dimensional settings in $\mathbb C^2$ and $\mathbb P^2(\mathbb C)$ (Astorg et al., 30 Mar 2026, Bianchi, 2016). A further line of work studies an analogue in parameter space, where near-parabolic perturbation enriches the connectedness locus of cubic families (Zhang, 22 Aug 2025).

1. Classical one-dimensional framework

In the classical autonomous setting, parabolic implosion concerns a holomorphic germ

$f(w)=w+w^2+O(w^3)$

with a simple parabolic fixed point at $0$, perturbed by

$f_\varepsilon(w)=f(w)+\varepsilon^2.$

The relevant asymptotic regime is the Lavaurs scale, recalled in the form

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$

under which the renormalized dynamics converges locally uniformly on compact subsets of the parabolic basin to a Lavaurs map $L_\sigma$ (Astorg et al., 29 Mar 2026).

The basic geometric objects are attracting and repelling petals, the parabolic basin

$\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$

and Fatou coordinates

$f_\varepsilon(z)=f(z)+\varepsilon^2$0

With the normalization

$f_\varepsilon(z)=f(z)+\varepsilon^2$1

the associated Lavaurs family is

$f_\varepsilon(z)=f(z)+\varepsilon^2$2

In this formulation, parabolic implosion is the appearance of a limiting transit map through the parabolic point after a perturbation has split the parabolic fixed point into two nearby simple fixed points (Astorg et al., 29 Mar 2026).

This classical picture also underlies the parameter-space formulation. For the quadratic family $f_\varepsilon(z)=f(z)+\varepsilon^2$3, Lavaurs proved that if

$f_\varepsilon(z)=f(z)+\varepsilon^2$4

then

$f_\varepsilon(z)=f(z)+\varepsilon^2$5

uniformly on compact subsets of the parabolic basin of $f_\varepsilon(z)=f(z)+\varepsilon^2$6, and the perturbed filled Julia sets satisfy

$f_\varepsilon(z)=f(z)+\varepsilon^2$7

(Zhang, 22 Aug 2025). This dynamical enrichment is the prototype for later work in non-autonomous and higher-dimensional settings.

2. Non-autonomous one-dimensional implosion

The non-autonomous theory replaces iteration of a single perturbed map by a time-dependent sequence. In the additive case, the fundamental system is

$f_\varepsilon(z)=f(z)+\varepsilon^2$8

so the map changes with the iterate index $f_\varepsilon(z)=f(z)+\varepsilon^2$9 (Astorg et al., 29 Mar 2026). The main theorem identifies a precise non-autonomous Lavaurs-type condition: $\mathbb C^2$0 with uniformly bounded $\mathbb C^2$1, and shows that the $\mathbb C^2$2-step dynamics is still asymptotically described by a classical Lavaurs map. More precisely,

$\mathbb C^2$3

locally uniformly on compact subsets of $\mathbb C^2$4, where the phase is the explicit weighted average

$\mathbb C^2$5

(Astorg et al., 29 Mar 2026). If $\mathbb C^2$6, then

$\mathbb C^2$7

locally uniformly on compact subsets of $\mathbb C^2$8 (Astorg et al., 29 Mar 2026).

This identifies a universal weighting kernel $\mathbb C^2$9: the whole perturbation history collapses to one scalar phase, but not to the unweighted arithmetic mean. The paper further gives a symmetric deterministic corollary, using $\mathbb P^2(\mathbb C)$0, and a random corollary: if $\mathbb P^2(\mathbb C)$1 is a bounded random sequence with

$\mathbb P^2(\mathbb C)$2

then

$\mathbb P^2(\mathbb C)$3

(Astorg et al., 29 Mar 2026). An ergodic skew-product version yields phase selection by the ergodic average $\mathbb P^2(\mathbb C)$4.

The proof is organized through the “eggbeater” decomposition into entering, passing through, and exiting the central region. Approximate fixed points

$\mathbb P^2(\mathbb C)$5

and the coordinate

$\mathbb P^2(\mathbb C)$6

produce an approximate translation dynamics; a correction

$\mathbb P^2(\mathbb C)$7

removes the cubic drift (Astorg et al., 29 Mar 2026). Summing the perturbative term $\mathbb P^2(\mathbb C)$8 along the orbit yields the weighted phase $\mathbb P^2(\mathbb C)$9.

A major conceptual point is that the non-autonomous additive theory is genuinely broader than the autonomous one. In the autonomous case a single scalar parameter determines the phase, whereas here the phase is an explicit functional of the whole perturbation sequence. At the same time, the limit remains a classical Lavaurs map rather than a new non-autonomous object (Astorg et al., 29 Mar 2026).

3. Möbius models, orthogonal polynomials, and explicit non-autonomous recurrences

A complementary approach replaces a general parabolic germ by the explicit Möbius map

$f(w)=w+w^2+O(w^3)$0

which has a parabolic fixed point at $f(w)=w+w^2+O(w^3)$1 since

$f(w)=w+w^2+O(w^3)$2

(Huneycutt et al., 22 Jan 2026). In this model, earlier work of Vivas showed that the associated Lavaurs map is the identity, so parabolic implosion takes the form of convergence to $f(w)=w+w^2+O(w^3)$3. The perturbed maps are

$f(w)=w+w^2+O(w^3)$4

and the non-autonomous composition is

$f(w)=w+w^2+O(w^3)$5

(Huneycutt et al., 22 Jan 2026).

The key reduction comes from the fact that each $f(w)=w+w^2+O(w^3)$6 is Möbius, so composition corresponds to multiplication of $f(w)=w+w^2+O(w^3)$7 matrices. The $f(w)=w+w^2+O(w^3)$8-step map has the form

$f(w)=w+w^2+O(w^3)$9

with coefficients expressed through recurrence sequences $0$0: $0$1 where

$0$2

$0$3

Thus

$0$4

follows from

$0$5

(Huneycutt et al., 22 Jan 2026).

The orthogonal-polynomial viewpoint is central. In the additive regime $0$6, the recurrence becomes

$0$7

and in the constant-$0$8 case this is exactly the Chebyshev recurrence (Huneycutt et al., 22 Jan 2026). In the multiplicative regime, the reference sequence is

$0$9

with the exact identities

$f_\varepsilon(w)=f(w)+\varepsilon^2.$0

These identities encode the finite-time cancellation mechanism behind the unperturbed limit $f_\varepsilon(w)=f(w)+\varepsilon^2.$1 (Huneycutt et al., 22 Jan 2026).

The main deterministic perturbative tool is the difference formula

$f_\varepsilon(w)=f(w)+\varepsilon^2.$2

with

$f_\varepsilon(w)=f(w)+\varepsilon^2.$3

(Huneycutt et al., 22 Jan 2026). It shows directly how stepwise errors accumulate. Since $f_\varepsilon(w)=f(w)+\varepsilon^2.$4 may be $f_\varepsilon(w)=f(w)+\varepsilon^2.$5, perturbations of size $f_\varepsilon(w)=f(w)+\varepsilon^2.$6 are critical: they can contribute $f_\varepsilon(w)=f(w)+\varepsilon^2.$7 after summation over $f_\varepsilon(w)=f(w)+\varepsilon^2.$8 terms. This is why cubic-scale perturbations $f_\varepsilon(w)=f(w)+\varepsilon^2.$9 are automatically small enough, whereas quadratic-scale perturbations require explicit cancellation conditions or random averaging (Huneycutt et al., 22 Jan 2026).

The paper proves several non-autonomous implosion theorems. For multiplicative perturbations

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$0

one has

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$1

uniformly on compact subsets of $$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$2 under any of the conditions

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$3

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$4

or

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$5

(Huneycutt et al., 22 Jan 2026). For combined multiplicative and additive perturbations the same convergence holds if $$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$6. For additive-only perturbations, the critical scale is

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$7

or

$$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$8

again yielding convergence to $$\varepsilon_n=\frac{\pi}{n}+\frac{\pi \sigma}{n^2}+O\!\Big(\frac{1}{n^{2+\alpha}}\Big),\qquad \alpha>0,$$9 uniformly on compact subsets (Huneycutt et al., 22 Jan 2026). In the random additive regime

$L_\sigma$0

with independent, bounded, mean-zero $L_\sigma$1, the convergence holds almost surely (Huneycutt et al., 22 Jan 2026).

An important structural caveat is that additive and multiplicative perturbations are not interchangeable even when the individual maps are conjugate. Theorem C produces sequences $L_\sigma$2 such that each

$L_\sigma$3

is conjugate to

$L_\sigma$4

yet

$L_\sigma$5

while

$L_\sigma$6

does not converge to the identity at any nonzero point (Huneycutt et al., 22 Jan 2026). This demonstrates that stepwise conjugacy does not preserve implosion asymptotics in the non-autonomous setting because the conjugacies vary with $L_\sigma$7.

4. Higher-dimensional parabolic implosion

In $L_\sigma$8, parabolic implosion occurs for maps tangent to the identity, where the local dynamics is anisotropic and involves characteristic directions, directors, invariant curves, and sectorial Fatou coordinates. A first two-dimensional Lavaurs theorem was obtained for a class of endomorphisms of $L_\sigma$9 tangent to the identity, with unperturbed normal form

$\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$0

where $\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$1, and perturbation

$\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$2

(Bianchi, 2016). If $\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$3 is an $\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$4-sequence,

$\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$5

then, near compact subsets of the invariant line $\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$6 in the attracting basin, subsequences of $\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$7 converge locally uniformly to a holomorphic transition map $\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$8 satisfying

$\mathcal B_f=\bigcup_n f^{-n}(P^\iota),$9

(Bianchi, 2016). This scalar Fatou-coordinate semiconjugacy is the two-dimensional analogue of the one-variable Lavaurs relation.

The two-dimensional theory already has global consequences. The paper defines the Lavaurs-Julia set

$f_\varepsilon(z)=f(z)+\varepsilon^2$00

and proves

$f_\varepsilon(z)=f(z)+\varepsilon^2$01

For regular polynomial endomorphisms it also proves

$f_\varepsilon(z)=f(z)+\varepsilon^2$02

hence discontinuity of the filled Julia set (Bianchi, 2016). The mechanism is that the limiting dynamics is no longer generated only by $f_\varepsilon(z)=f(z)+\varepsilon^2$03, but by $f_\varepsilon(z)=f(z)+\varepsilon^2$04 together with the hidden transition $f_\varepsilon(z)=f(z)+\varepsilon^2$05.

A more recent extension develops parabolic implosion in complex dimension $f_\varepsilon(z)=f(z)+\varepsilon^2$06 for holomorphic germs tangent to the identity at order $f_\varepsilon(z)=f(z)+\varepsilon^2$07, under the hypothesis of a non-degenerate characteristic direction with a formal non-singular invariant curve and director $f_\varepsilon(z)=f(z)+\varepsilon^2$08 satisfying

$f_\varepsilon(z)=f(z)+\varepsilon^2$09

(Astorg et al., 30 Mar 2026). In adapted coordinates the perturbation has the normal form

$f_\varepsilon(z)=f(z)+\varepsilon^2$10

with

$f_\varepsilon(z)=f(z)+\varepsilon^2$11

The unperturbed map admits incoming and outgoing Fatou coordinates

$f_\varepsilon(z)=f(z)+\varepsilon^2$12

with asymptotics

$f_\varepsilon(z)=f(z)+\varepsilon^2$13

$f_\varepsilon(z)=f(z)+\varepsilon^2$14

(Astorg et al., 30 Mar 2026).

The resulting limit map is genuinely two-dimensional. In Fatou coordinates, the transit is the affine map

$f_\varepsilon(z)=f(z)+\varepsilon^2$15

and the associated Lavaurs map is

$f_\varepsilon(z)=f(z)+\varepsilon^2$16

For perturbation sequences satisfying

$f_\varepsilon(z)=f(z)+\varepsilon^2$17

the main convergence theorem gives

$f_\varepsilon(z)=f(z)+\varepsilon^2$18

uniformly on compact subsets of the parabolic basin (Astorg et al., 30 Mar 2026). The new feature relative to one dimension is the transverse multiplier $f_\varepsilon(z)=f(z)+\varepsilon^2$19, so the limit is a translation-plus-scaling rather than a pure translation.

These results also have consequences for endomorphisms of $f_\varepsilon(z)=f(z)+\varepsilon^2$20. Under suitable assumptions, the paper proves

$f_\varepsilon(z)=f(z)+\varepsilon^2$21

and, assuming $f_\varepsilon(z)=f(z)+\varepsilon^2$22, discontinuity of $f_\varepsilon(z)=f(z)+\varepsilon^2$23 at the unperturbed map (Astorg et al., 30 Mar 2026). A related additive non-autonomous theory for fibered holomorphic endomorphisms of $f_\varepsilon(z)=f(z)+\varepsilon^2$24 yields

$f_\varepsilon(z)=f(z)+\varepsilon^2$25

for $f_\varepsilon(z)=f(z)+\varepsilon^2$26-almost every base point $f_\varepsilon(z)=f(z)+\varepsilon^2$27, and under a non-collinearity condition on three periodic orbit averages, the strong discontinuity statement

$f_\varepsilon(z)=f(z)+\varepsilon^2$28

for some nonempty open set $f_\varepsilon(z)=f(z)+\varepsilon^2$29 (Astorg et al., 29 Mar 2026).

The higher-dimensional theory is therefore not a formal lifting of the one-dimensional one. It retains the incoming/outgoing Fatou-coordinate structure, but characteristic directions, directors, invariant curves, and transverse scaling become essential.

5. Parameter-space implosion

Parabolic implosion also has a parameter-space form. For the cubic family

$f_\varepsilon(z)=f(z)+\varepsilon^2$30

the parabolic slice $f_\varepsilon(z)=f(z)+\varepsilon^2$31 consists of maps

$f_\varepsilon(z)=f(z)+\varepsilon^2$32

with degenerate parabolic parameter $f_\varepsilon(z)=f(z)+\varepsilon^2$33, where

$f_\varepsilon(z)=f(z)+\varepsilon^2$34

(Zhang, 22 Aug 2025). The perturbation regime is

$f_\varepsilon(z)=f(z)+\varepsilon^2$35

with $f_\varepsilon(z)=f(z)+\varepsilon^2$36 and $f_\varepsilon(z)=f(z)+\varepsilon^2$37 automatically under $f_\varepsilon(z)=f(z)+\varepsilon^2$38 (Zhang, 22 Aug 2025).

The central object is the central part $f_\varepsilon(z)=f(z)+\varepsilon^2$39 of the cubic connectedness locus, and the core implosion statement is

$f_\varepsilon(z)=f(z)+\varepsilon^2$40

where

$f_\varepsilon(z)=f(z)+\varepsilon^2$41

is the parameter escaping locus defined through the cubic Lavaurs map (Zhang, 22 Aug 2025). Thus the upper limit of nearby cubic loci is strictly smaller than the parabolic locus, and the “missing” part is organized by Lavaurs dynamics.

The paper introduces the alternative family

$f_\varepsilon(z)=f(z)+\varepsilon^2$42

whose critical points are explicitly marked at $f_\varepsilon(z)=f(z)+\varepsilon^2$43 and $f_\varepsilon(z)=f(z)+\varepsilon^2$44, and which is conjugate to the parabolic cubic family through

$f_\varepsilon(z)=f(z)+\varepsilon^2$45

(Zhang, 22 Aug 2025). The stable parabolic locus in the $f_\varepsilon(z)=f(z)+\varepsilon^2$46-plane has two components $f_\varepsilon(z)=f(z)+\varepsilon^2$47 and $f_\varepsilon(z)=f(z)+\varepsilon^2$48; $f_\varepsilon(z)=f(z)+\varepsilon^2$49 is a Jordan curve, and Misiurewicz parabolic parameters are dense on $f_\varepsilon(z)=f(z)+\varepsilon^2$50 (Zhang, 22 Aug 2025).

The main topological theorem concerns connected components of the first-level escaping locus

$f_\varepsilon(z)=f(z)+\varepsilon^2$51

Each connected component $f_\varepsilon(z)=f(z)+\varepsilon^2$52 admits a conformal parametrization

$f_\varepsilon(z)=f(z)+\varepsilon^2$53

such that

$f_\varepsilon(z)=f(z)+\varepsilon^2$54

where $f_\varepsilon(z)=f(z)+\varepsilon^2$55 or $f_\varepsilon(z)=f(z)+\varepsilon^2$56 is the escaping critical point (Zhang, 22 Aug 2025). Under the additional assumption

$f_\varepsilon(z)=f(z)+\varepsilon^2$57

each first-level component is a croissant, its boundary meets $f_\varepsilon(z)=f(z)+\varepsilon^2$58 in exactly one point, and distinct components have disjoint closures unless both are attached at $f_\varepsilon(z)=f(z)+\varepsilon^2$59 (Zhang, 22 Aug 2025). The attachment point is either a Misiurewicz parabolic parameter or the degenerate parameter $f_\varepsilon(z)=f(z)+\varepsilon^2$60; if it is Misiurewicz parabolic, the attached escaping component is unique, while exactly two components are attached at $f_\varepsilon(z)=f(z)+\varepsilon^2$61 (Zhang, 22 Aug 2025).

This parameter-space theory is explicitly presented as a counterpart to Lavaurs’ dynamical implosion for quadratic Julia sets. The paper does not prove a full global conjugacy between the cubic parameter escaping locus and the quadratic Lavaurs escaping region, but it establishes the first topological ingredients: half-plane uniformization, unique attachment points, and croissant geometry (Zhang, 22 Aug 2025).

6. Mechanisms, scope, and limitations

Across these settings, parabolic implosion is driven by a common mechanism: perturbations on a critical scale create a long transit through a near-parabolic gate, and the cumulative effect of $f_\varepsilon(z)=f(z)+\varepsilon^2$62 or $f_\varepsilon(z)=f(z)+\varepsilon^2$63 small errors produces a finite limit. In the additive one-dimensional theory, the leading scale is

$f_\varepsilon(z)=f(z)+\varepsilon^2$64

with second-order corrections $f_\varepsilon(z)=f(z)+\varepsilon^2$65 encoded by $f_\varepsilon(z)=f(z)+\varepsilon^2$66 and weighted by $f_\varepsilon(z)=f(z)+\varepsilon^2$67 (Astorg et al., 29 Mar 2026). In the Möbius model, the multiplicative critical scale is

$f_\varepsilon(z)=f(z)+\varepsilon^2$68

while the additive critical scale is

$f_\varepsilon(z)=f(z)+\varepsilon^2$69

(Huneycutt et al., 22 Jan 2026). In dimension $f_\varepsilon(z)=f(z)+\varepsilon^2$70, the scaling

$f_\varepsilon(z)=f(z)+\varepsilon^2$71

again governs the transit, but the limiting map includes a transverse factor $f_\varepsilon(z)=f(z)+\varepsilon^2$72 (Astorg et al., 30 Mar 2026).

Several recurrent misconceptions are explicitly ruled out by the recent literature. First, the non-autonomous theory is not merely a reformulation of the autonomous one: the perturbation history matters, and in the general additive theory it survives precisely through the weighted phase $f_\varepsilon(z)=f(z)+\varepsilon^2$73 (Astorg et al., 29 Mar 2026). Second, individual stepwise conjugacies do not determine non-autonomous implosion asymptotics, as shown by the Möbius model where conjugate additive and multiplicative steps can have different limiting compositions (Huneycutt et al., 22 Jan 2026). Third, the Möbius-based identity limit should not be confused with the full Douady–Lavaurs theory: the model captures the accumulated near-parabolic transition mechanism, but the paper states explicitly that it does not build explicit Fatou coordinates or a limiting translation flow, and that the analogy is structural rather than full-strength holomorphic-dynamical (Huneycutt et al., 22 Jan 2026).

The main limitations are also explicit. The Möbius model is deliberately local and model-based: no horn maps are analyzed, the limiting map is the identity because of the special choice $f_\varepsilon(z)=f(z)+\varepsilon^2$74, and the results do not provide a complete nonlinear classification (Huneycutt et al., 22 Jan 2026). The general additive non-autonomous theory treats additive perturbations of arbitrary parabolic germs, but its stated scope is additive rather than multiplicative or fully mixed (Astorg et al., 29 Mar 2026). In dimension $f_\varepsilon(z)=f(z)+\varepsilon^2$75, the hypothesis

$f_\varepsilon(z)=f(z)+\varepsilon^2$76

equivalently $f_\varepsilon(z)=f(z)+\varepsilon^2$77, is identified as the threshold that makes the error terms summable in the eggbeater analysis; the paper notes that this is stronger than in earlier work and does not yet reach the full expected range of directors (Astorg et al., 30 Mar 2026). In parameter space, the cubic theory establishes the topology of escaping components and their relation to Lavaurs dynamics, but it does not construct the full conjectural parameter-dynamical double covering between cubic parameter implosion and the quadratic Lavaurs escaping region (Zhang, 22 Aug 2025).

Taken together, these results show that parabolic implosion is not a single theorem but a family of asymptotic transition phenomena. In one variable it is the emergence of Lavaurs maps from near-parabolic perturbations; in non-autonomous settings it becomes a weighted memory effect of the perturbation sequence; in higher dimension it acquires transverse moduli and characteristic-direction geometry; and in parameter space it appears as the enrichment of connectedness loci by croissant-shaped escaping components organized by Lavaurs dynamics (Astorg et al., 29 Mar 2026, Huneycutt et al., 22 Jan 2026, Astorg et al., 30 Mar 2026, Zhang, 22 Aug 2025, Bianchi, 2016).