Polynomial Skew Products Dynamics

• Polynomial skew products are holomorphic maps of C² that split dynamics between an autonomous base and fiberwise evolving polynomials.
• They integrate one-variable polynomial dynamics, pluripotential theory, and bifurcation theory to enable explicit multidimensional analysis.
• Their structured dynamics facilitate the classification of invariant fibers, Fatou components, and hyperbolicity regimes, yielding rich topological insights.

Polynomial skew products are holomorphic maps of the form

$$f:\mathbb C^2\to\mathbb C^2,\qquad f(z,w)=(p(z),q(z,w)),$$

in which the first coordinate evolves autonomously and the second coordinate evolves fiberwise along the orbit of the base point. This split structure makes them a canonical class of two-variable holomorphic dynamical systems: they preserve vertical fibers, admit both local and global compactifications in many regular settings, and support a theory that combines one-variable polynomial dynamics, pluripotential theory, hyperbolic dynamics, and bifurcation theory (Roeder, 2010, Ueno, 2012, Astorg et al., 2018).

1. Basic form, iterates, and regular extensions

A polynomial skew product is specified by a base polynomial $p$ and a family of fiber polynomials $q_z(w):=q(z,w)$. Its iterates retain the same triangular form: $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$ Accordingly, the derivative matrix is triangular,

$$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$

so the critical set is controlled by the base critical points and the fiber-critical equation $\partial_w q(z,w)=0$ (Nakane, 2012).

A central global subclass is formed by regular polynomial skew products, namely skew products of common degree $d$ that extend holomorphically to $\mathbb P^2$. In the normalized setting $p(z)=z^d+O(z^{d-1})$ and $q(z,w)=w^d+O_z(w^{d-1})$, the extension takes the homogeneous form

$p$0

The induced dynamics on the line at infinity $p$1 is then governed by the one-variable polynomial

$p$2

which supplies a third one-dimensional dynamical system, in addition to the base map $p$3 and the fiber maps $p$4 (Roeder, 2010).

This three-level structure—base, fibers, and infinity—organizes much of the subject. A plausible implication is that polynomial skew products occupy an intermediate position between one-variable dynamics and general holomorphic endomorphisms of $p$5: they retain enough rigidity for explicit analysis while already exhibiting genuinely higher-dimensional behavior.

2. Invariant fibers and the local classification of dynamics

An invariant fiber is a vertical line $p$6 with $p$7. On such a fiber, the skew product restricts to the one-variable polynomial

$p$8

and the local two-dimensional dynamics is organized by the multiplier $p$9. The standard classification is: super-attracting if $q_z(w):=q(z,w)$0, geometrically attracting if $q_z(w):=q(z,w)$1, parabolic if $q_z(w):=q(z,w)$2 is a root of unity, and elliptic if $q_z(w):=q(z,w)$3 with $q_z(w):=q(z,w)$4 and $q_z(w):=q(z,w)$5 (Peters et al., 2016).

A basic notion in this setting is a bulging Fatou component: a two-dimensional Fatou component whose slice over the invariant fiber coincides with a one-variable Fatou component of $q_z(w):=q(z,w)$6, and whose nearby vertical slices provide its continuation. This notion expresses the idea that nearby two-dimensional normality is inherited from the one-dimensional restriction.

The elliptic case is governed by the small-divisor problem for the base linearization. If the multiplier is Brjuno, so that $q_z(w):=q(z,w)$7, and every critical point of $q_z(w):=q(z,w)$8 lies in the basin of an attracting or parabolic cycle, then the local dynamics near the elliptic invariant fiber is completely described by the fiber map: every Fatou component of $q_z(w):=q(z,w)$9 bulges, and in a neighborhood of the invariant fiber the only Fatou components are these bulging components. In particular, there are no wandering Fatou components there. The proof uses local normalization near parabolic cycles in the fiber and a family of expanding fiberwise metrics $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$0 for which the vertical derivative expands uniformly (Peters et al., 2016).

The Brjuno hypothesis is not merely technical. In the non-Brjuno, Cremer case, the normalization may fail because the cohomological equations defining invariant holomorphic curves do not converge. Explicit examples such as

$$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$1

show that a holomorphic invariant curve $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$2 need not exist, and parabolic basins of the fiber map need not bulge (Peters et al., 2016).

3. Fatou components and the wandering-domain problem

The classification of Fatou components near invariant fibers proceeds differently in the super-attracting, attracting, elliptic, and parabolic regimes. Near a super-attracting invariant fiber, the known picture is rigid: the only two-dimensional Fatou components are bulging components coming from the fiber restriction, and there are no wandering domains in that neighborhood (Peters et al., 2016).

For geometrically attracting fibers, several non-wandering results are available under different hypotheses. In the attracting skew-product setting with $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$3, $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$4, and subhyperbolic base polynomial $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$5, there are no wandering Fatou components contained in the attracting basin of the invariant fiber; the proof uses expansion on $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$6, bounds on the escape of critical orbits in almost all fibers, and a linearization map along unstable manifolds (Peters et al., 2015). A different approach shows that if the base multiplier is sufficiently small, then every Fatou component iterates to a bulging Fatou component, every forward orbit of a vertical Fatou disk intersects a bulging component, and wandering Fatou components do not occur (Ji, 2018).

More specialized rigidity results hold in normal forms. In the unicritical attracting model

$$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$7

every Fatou component is an extension of a Fatou component of the one-variable polynomial $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$8; in particular there is no wandering Fatou component in the basin of the invariant line (Ji et al., 2022). Under non-uniform hyperbolicity assumptions on the restriction $$f^n(z,w)=\bigl(p^n(z),\,Q_z^n(w)\bigr),\qquad Q_z^n=q_{p^{n-1}(z)}\circ\cdots\circ q_z.$$9—either Topological Collet–Eckmann plus Weak Regularity, or positive Lyapunov exponents at critical values with no parabolic cycles—the Fatou set in the basin of the invariant line coincides with the union of the basins of attracting cycles, the Julia set there has Lebesgue measure zero, and wandering Fatou components are excluded (Ji, 2019).

The parabolic case is qualitatively different. Polynomial skew products with a parabolic invariant fiber can possess wandering Fatou components, and explicit examples are given by skew products of the form

$$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$0

with $$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$1 and $$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$2. The construction uses parabolic implosion, approximate Fatou coordinates, and the convergence

$$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$3

on the product of parabolic basins, where $$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$4 is a Lavaurs map. This yields the first polynomial endomorphisms of $$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$5 with wandering Fatou components, and the examples can be chosen to extend holomorphically to $$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$6 (Astorg et al., 2014).

4. Green functions, hyperbolicity, and topological dichotomies

Polynomial skew products admit several Green functions reflecting the split dynamics. The base Green function is

$$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$7

Fiberwise one defines

$$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$8

when the limit exists, and weighted Green functions $$Df(z,w)= \begin{pmatrix} p'(z) & 0\ \partial_z q(z,w) & \partial_w q(z,w) \end{pmatrix},$$9 or $\partial_w q(z,w)=0$0 are introduced to balance the horizontal and vertical growth when $\partial_w q(z,w)=0$1 and $\partial_w q(z,w)=0$2 differ. The relevant weight is determined by the leading $\partial_w q(z,w)=0$3-degree $\partial_w q(z,w)=0$4 of the coefficient $\partial_w q(z,w)=0$5 of $\partial_w q(z,w)=0$6 and by the comparison of $\partial_w q(z,w)=0$7 and $\partial_w q(z,w)=0$8. In the case $\partial_w q(z,w)=0$9, the special weight $d$0 gives a refined Green function

$d$1

which is continuous and plurisubharmonic on $d$2 and pluriharmonic on a forward-invariant region $d$3 (Ueno, 2012).

In the regular Axiom A setting, Jonsson’s criterion characterizes hyperbolicity in terms of base hyperbolicity and vertical expansion over $d$4 and over the attracting cycles of $d$5. The associated second Julia set is

$d$6

and the postcritical sets over $d$7, $d$8, and the line at infinity govern whether the skew product is Axiom A (Roeder, 2010, Nakane, 2012).

These ingredients lead to strong topological consequences. For an Axiom A polynomial skew product that extends holomorphically to $d$9, if the map is connected in Jonsson’s sense—$\mathbb P^2$0 connected and $\mathbb P^2$1 connected for all $\mathbb P^2$2—then every Fatou component is homeomorphic to an open ball. If connectedness fails, then some Fatou component has infinitely generated first homology (Roeder, 2010). A finer postcritical analysis decomposes the saddle part of the nonwandering set into basic sets $\mathbb P^2$3, studies the accumulation sets of the fiber-critical set $\mathbb P^2$4, and characterizes when pointwise, component-wise, and global postcritical accumulation coincide in terms of the strata

$\mathbb P^2$5

and their closedness or openness properties (Nakane, 2012).

5. Parameter spaces, bifurcation currents, and robust instability

For holomorphic families of polynomial skew products, the bifurcation current is defined by

$\mathbb P^2$6

where $\mathbb P^2$7 is the sum of Lyapunov exponents. In skew-product families with fixed base, the Lyapunov function decomposes into a base part and a vertical part, and the vertical bifurcation current admits a fiberwise decomposition over $\mathbb P^2$8. In the quadratic family

$\mathbb P^2$9

the accumulation of the bifurcation locus on the hyperplane at infinity is described exactly by the set

$p(z)=z^d+O(z^{d-1})$0

and the intersection formula

$p(z)=z^d+O(z^{d-1})$1

records the distribution of bifurcations near infinity (Astorg et al., 2018).

The same parametric framework yields a complete classification of the hyperbolic components with access to infinity in the escape locus, via a combinatorial datum $p(z)=z^d+O(z^{d-1})$2 recording how the roots of $p(z)=z^d+O(z^{d-1})$3 distribute among the bounded Fatou components of $p(z)=z^d+O(z^{d-1})$4. Stability in the sense of holomorphic motions of repelling cycles preserves vertical expansion, and therefore preserves hyperbolicity inside stable components of skew-product families (Astorg et al., 2018).

Higher-dimensional bifurcation phenomena are substantially richer than in one variable. If the base polynomial has Julia set not totally disconnected and is neither a Chebyshev polynomial nor a power map, then near any bifurcation parameter one can find parameters where $p(z)=z^d+O(z^{d-1})$5 critical points bifurcate independently, for every $p(z)=z^d+O(z^{d-1})$6 up to the dimension of the parameter space. Consequently,

$p(z)=z^d+O(z^{d-1})$7

the support of the bifurcation measure has non-empty interior, and its Hausdorff dimension is maximal at every point of its support (Astorg et al., 2020).

A complementary mechanism for robust instability is provided by holomorphic blenders. If $p(z)=z^d+O(z^{d-1})$8 bifurcates, then the product map $p(z)=z^d+O(z^{d-1})$9 can be approximated by polynomial skew products whose suitable iterates possess blenders of repelling type or saddle type. As a consequence, such product maps belong to the closure of the interior of the bifurcation locus in $q(z,w)=w^d+O_z(w^{d-1})$0, and also to the closure of the interior of the set of endomorphisms having an attracting set of non-empty interior (Taflin, 2017).

6. Rigidity, asymptotic models, and recent directions

Several recent results identify rigid subclasses of polynomial skew products. A regular polynomial skew product is called special if it is triangularly conjugate either to a product $q(z,w)=w^d+O_z(w^{d-1})$1 with $q(z,w)=w^d+O_z(w^{d-1})$2 power maps or $q(z,w)=w^d+O_z(w^{d-1})$3Chebyshev maps, or to

$q(z,w)=w^d+O_z(w^{d-1})$4

where $q(z,w)=w^d+O_z(w^{d-1})$5 is the Dickson polynomial. Specialness is equivalent to semiconjugacy to an affine self-map in skew-product form on a two-dimensional connected commutative algebraic group, and also equivalent to the condition that all multipliers of the skew product lie in a fixed number field (Zhang, 14 Apr 2026).

A different rigidity problem concerns symmetries of the Julia set. For polynomial skew products whose Julia sets have infinitely many symmetries, the map is birationally conjugate to a rotational product, and in normal form this happens exactly in four families: $q(z,w)=w^d+O_z(w^{d-1})$6, $q(z,w)=w^d+O_z(w^{d-1})$7, $q(z,w)=w^d+O_z(w^{d-1})$8, and semiconjugate forms $q(z,w)=w^d+O_z(w^{d-1})$9 arising from

$p$00

The corresponding symmetry groups are subgroups of $p$01 determined by the centroids of the base and fiber maps (Ueno, 2012).

Near infinity, the dominant asymptotic model can be read off from a reverse Newton polygon of $p$02. The vertices $p$03 and the intercepts $p$04 determine a dominant term $p$05, a monomial model

$p$06

and an invariant horn region $p$07 on which $p$08 is conjugate to $p$09 by a Böttcher coordinate $p$10 satisfying $p$11. Depending on the comparison of $p$12 with the $p$13, the invariant region is a lower horn, an upper horn, or a double horn, and it lies in the attracting basin of a fixed or indeterminacy point at infinity, or in the closure of the attracting basins of two points at infinity (Ueno, 2024).

Local skew-product structure also appears in superattracting germs with small relative degree $p$14. Such germs are formally conjugated to

$p$15

with $p$16 of degree $p$17, which induces dynamics on the Berkovich affine line over $p$18. In that setting there is an invariant compact set $p$19 supporting a natural mixing invariant measure, and under a non-recurrence assumption on critical branches the associated positive closed current $p$20 admits a geometric representation as an average of integration currents over curves in $p$21; in particular, $p$22 is uniformly laminar outside the origin (Dujardin et al., 12 Jul 2025).

Finally, the Axiom A subclass now admits an algorithmic theory. For an Axiom A polynomial skew product, the chain recurrent set, the non-wandering set, and the closure of the periodic orbits coincide and are computable; the algorithm also separates the expanding, attracting, and saddle-type basic sets. As consequences, Axiom A is semi-decidable on the closure of the Axiom A locus, and the Axiom A locus in fixed degree is lower semi-computable (Boyd et al., 11 Aug 2025).