Postcritically Finite Rational Maps

• Postcritically Finite Rational Maps are rational functions defined on the Riemann sphere with finite postcritical sets, ensuring each critical point has a periodic or preperiodic orbit.
• They underpin advanced holomorphic dynamics by enabling canonical decompositions, combinatorial models, and algorithmic classification methods.
• They play a crucial role in rigidity theory and the study of complex dynamics, unifying families such as polynomials, Lattès maps, and Newton maps.

A postcritically finite (PCF) rational map is a rational function $$f:\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}$$ of degree at least two on the Riemann sphere $\widehat{\mathbb{C}}$, for which the postcritical set $P(f)=\bigcup_{n\geq1}f^n(\mathrm{Crit}(f))$ is finite; that is, every critical point has a strictly preperiodic or periodic forward orbit. Such maps are central to holomorphic dynamics, where combinatorial, topological, and geometric models for their global dynamics have been developed, including Thurston equivalence, graph invariants, and canonical decompositions. The structure of PCF rational maps forms the basis for substantial progress in classification, rigidity theory, and the geometric understanding of rational dynamics, unifying polynomials, Lattès maps, and Newton maps under a common dynamical theme.

1. Definitions and Basic Properties

Let $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be a rational map of degree $d\geq2$. The critical set is $$\mathrm{Crit}(f)=\{z\in\widehat{\mathbb{C}}\,:\,f'(z)=0\}$$, and the postcritical set is

$P(f)=\bigcup_{n\ge1}f^n(\mathrm{Crit}(f)),$

where $f^n$ is the $n$-th iterate. $f$ is postcritically finite (PCF) if $\#P(f)<\infty$. The Fatou set $F(f)$ is the maximal open set where the family $\{f^n\}$ is normal; its complement $J(f)=\widehat{\mathbb{C}}\setminus F(f)$ is the Julia set. For PCF maps with nonempty Fatou set, all Fatou components are simply connected and extend continuously to their boundaries, and $J(f)$ is locally connected (Gao et al., 2015, Dudko et al., 2022).

PCF property admits both complex and $p$-adic analogs (Benedetto et al., 2012, Benedetto et al., 2020). PCF maps include polynomials with all critical orbits finite, Lattès maps from torus endomorphisms, and specialized families such as Chebyshev and McMullen maps.

2. Canonical Decompositions: Multicurves, Block Types, and Expanding Quotients

A central tool for understanding PCF rational maps is the canonical decomposition along invariant multicurves. For a PCF rational map $f:(S^2,A)$, an invariant multicurve $\Gamma$ is a finite set of disjoint, essential simple closed curves in $S^2\setminus A$ invariant under $f^{-1}$ (Bonnot et al., 2010, Bonnot et al., 2011, Dudko et al., 2022).

The decomposition splits $S^2$ into pieces with first-return maps of three classical types:

• Homeomorphisms
• Torus maps
• Rational maps

Within rational type, the canonical crochet–Sierpiński decomposition (Dudko et al., 2022) applies: each piece (termed "small sphere") falls into one of two classes:

• Sierpiński type: Julia set is a Sierpiński carpet, every Fatou component is a Jordan disk with disjoint closure.
• Crochet (Newton-like) type: any two Fatou domains can be connected by a finite chain of Fatou components with intersecting boundaries. Equivalently, the sphere contains a forward-invariant zero-entropy graph through marked points (Dudko et al., 2022, Gao et al., 2015, Lodge et al., 2015).

The unique minimal multicurve $\Gamma_{\mathrm{cro}}$ is characterized equivalently as the boundary of maximal Sierpiński domains plus a generating set of "bicycles" (strongly connected multicurve classes). On the corresponding quotient, Sierpiński blocks become spheres and crochet blocks points; the quotient space is a finite cactoid supporting a totally expanding quotient dynamics. Any other expanding quotient factors through this maximal expanding quotient (Dudko et al., 2022).

3. Algorithmic and Combinatorial Models

Effective algorithms exist for computing the canonical block decomposition for a PCF rational map given by combinatorial data (such as a biset or critical orbit portrait) (Dudko et al., 2022, Floyd et al., 2021, Lodge et al., 2015). The process, termed the Crochet Algorithm, proceeds via:

• Touching-Equivalence Clustering: Fatou components are grouped if their closures intersect.
• Successive Multicurve Decomposition: After clustering, decompose along boundary multicurves, analyze return maps, and classify each resulting small sphere as Sierpiński or crochet.
• Backward-Invariance and Merging of Crochet Unicycles: Recursing until all blocks are classified, and primitive crochet unicycles—whose removal would disconnect two crochet blocks—are merged.
• Polynomial-Time Termination: All steps (curve preimages, cluster connectivity, zero-entropy tests) are implementable in finite time and polynomial bit-size in the input model.

The combinatorial models extend to concrete finite graphs carrying the postcritical set and critical orbit data. In the polynomial or Newton map cases, abstract portraits or extended Newton graphs describe the dynamics up to combinatorial (Thurston) equivalence (Floyd et al., 2021, Lodge et al., 2015).

4. Topological and Metric Model Spaces

The quotient by collapsing crochet regions and annuli associated with bicycles produces a maximal expanding cactoid: a finite union of spheres (Sierpiński pieces) and intervals (edges/bicycles) meeting only at marked points (Dudko et al., 2022). The induced map on the cactoid is totally topologically expanding with metric expansion on each node or segment by a factor $\lambda>1$.

Metric models can be further generalized to Böttcher-expanding Thurston maps—branched covers of the sphere admitting a length-metric on $S^2\setminus A^\infty$ where all lifts are strictly contractive, and local Böttcher normalization at attracting cycles. The same canonical decomposition and cactoid quotient extends to this setting, giving a unifying metric framework modeling the PCF case (Dudko et al., 2022).

5. Dynamics, Rigidity, and Finiteness Phenomena

PCF rational maps exhibit strong rigidity properties:

• Thurston Equivalence: If a PCF rational map admits no multicurve obstruction, it is combinatorially equivalent to a unique rational map, up to Möbius conjugacy (Bonnot et al., 2010, Bonnot et al., 2011, Gao et al., 2015).
• Sierpiński Characterization: A PCF rational map with Julia set homeomorphic to the Sierpiński carpet is Thurston equivalent to an expanding Thurston map if and only if every Fatou component is a Jordan disk with disjoint closure (Gao et al., 2015).
• Global Moduli Discreteness and Height Bounds: The moduli space of PCF maps (excluding flexible Lattès cases) is discrete over finite fields and p-adic or complex moduli (Benedetto et al., 2012, Benedetto et al., 2020). For degree $d$ and fixed number field, only finitely many non-Lattès PCF rational maps exist.
• Parameter Rigidity: In any one-parameter analytic family (excluding isotrivial and flexible Lattès cases), PCF parameters are isolated in both complex and $p$-adic settings (Benedetto et al., 2020).
• Prescribed Portraits: Every finite set or abstract polynomial portrait is realizable as the postcritical set or ramification portrait of a PCF map, subject to explicit combinatorial constraints (DeMarco et al., 2017, Floyd et al., 2021).

6. Classification of Special Families

6.1 Newton Maps

Postcritically finite Newton maps $N_p(z)=z-p(z)/p'(z)$ admit a full combinatorial classification via extended Newton graphs—finite graphs (with Hubbard trees) whose vertices and edges encode critical and postcritical dynamics, subject to nine combinatorial axioms. Every such abstract graph comes from a unique PCF Newton map, and vice versa (Lodge et al., 2015).

6.2 Decomposition with Wandering Jordan Curves and Renormalization

Maps admitting a Cantor multicurve are further decomposable: Julia sets can contain uncountably many wandering Jordan curves (components whose forward orbits never repeat), with complements consisting of renormalizations (small filled Julia sets supporting polynomial-like return maps) and points. The quotient dynamics is modeled by a dendrite (tree-like continuum), and surgery procedures allow realizing any polynomial renormalizations and prescribed combinatorics (Cui et al., 2014).

7. Illustrative Examples and Applications

• Sierpiński Carpet Examples: McMullen maps $z^n+\lambda/z^m$ (for small $\lambda$) yield PCF rational maps with Sierpiński carpet Julia sets; Fatou components are Jordan disks with disjoint closure.
• Newton Map Examples: For $d\ge3$, the Newton map of $p(z)=z^d-1$ has a postcritically finite Newton map structure with an extended Newton graph equal to its channel diagram (Lodge et al., 2015).
• Hyperbolic Tree Mapping Schemes: For hyperbolic rational maps with finitely connected Fatou set (all periodic Fatou components are disks), rescaling limits are captured by irreducible postcritically finite hyperbolic tree mapping schemes, yielding abundant combinatorial models generalizing the PCF case (2004.02797).

PCF rational maps thus form a central organizing class in complex dynamics, unifying a vast swath of structural, rigidity, combinatorial, and metric phenomena, and connecting decomposition theory, moduli, portrait realization, and parameter arithmetic in a single framework (Bonnot et al., 2010, Bonnot et al., 2011, Cui et al., 2014, Gao et al., 2015, Lodge et al., 2015, DeMarco et al., 2017, 2004.02797, Benedetto et al., 2020, Floyd et al., 2021, Dudko et al., 2022).