Fixed-Point Marked Quasi-Blaschke Products

• Fixed-point-marked quasi-Blaschke products are hyperbolic rational maps whose Julia sets form quasi-circles and include an ordered marking of all fixed points.
• The moduli space QB_d^(fm) is constructed by quotienting the fixed-point marked space by PSL(2,C), enabling rigorous uniformization and multiplier coordinate analysis.
• Key analytic results include a biholomorphic simultaneous uniformization with genuine Blaschke products and a non-degeneracy theorem linking Weil–Petersson and pressure norms.

Fixed-point-marked quasi-Blaschke products form a rigidified moduli problem in one-dimensional complex dynamics. For degree $d\ge 2$, a quasi-Blaschke product is a hyperbolic rational map whose Julia set is a quasi-circle and which fixes each of the two Fatou components; the fixed-point-marked theory records, in addition, all $d+1$ fixed points in an ordered tuple and then passes to the quotient by $\PSL(2,\mathbb C)$. The resulting moduli space $QB_d^{fm}$ contains the subspace $B_d^{fm}$ of genuine Blaschke products, and its analytic theory is organized by two central results: a biholomorphic simultaneous uniformization $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$, and a non-degeneracy theorem for the Weil–Petersson, equivalently pressure, semi-norm away from the super-attracting locus $\mathcal{SA}_d^{fm}$ (He et al., 22 Jul 2025).

1. Definition of the objects and the moduli spaces

For $d\ge 2$, a degree-$d$ Blaschke product is a rational map $f:\mathbb P^1\to \mathbb P^1$ of the form

$d+1$0

where $d+1$1, $d+1$2, and $d+1$3. The fixed-point-marked ambient space is

$d+1$4

which is a smooth manifold. The group $d+1$5 acts by

$d+1$6

and this action is free on the hyperbolic locus (He et al., 22 Jul 2025).

A quasi-Blaschke product is defined as a hyperbolic rational map whose Julia set $d+1$7 is a quasi-circle and which fixes each of the two Fatou components. The connected component $d+1$8 of fixed-point-marked degree-$d+1$9 quasi-Blaschke products containing

$\PSL(2,\mathbb C)$0

gives, after quotienting by $\PSL(2,\mathbb C)$1,

$\PSL(2,\mathbb C)$2

The Blaschke locus is the subspace

$\PSL(2,\mathbb C)$3

Both $\PSL(2,\mathbb C)$4 and $\PSL(2,\mathbb C)$5 are complex manifolds.

The fixed-point marking is not auxiliary decoration. It removes residual Möbius symmetry by fixing reference points, so that the quotient acquires a usable moduli interpretation. In particular, the marked theory distinguishes the two attracting basins and orders the remaining fixed points on the Julia set, which is essential for the uniformization and metric results.

2. Standard representatives, marking conventions, and circle dynamics

A standard representative of a class $\PSL(2,\mathbb C)$6 in $\PSL(2,\mathbb C)$7 or $\PSL(2,\mathbb C)$8 is defined by the conditions that $\PSL(2,\mathbb C)$9 is a Blaschke product, $QB_d^{fm}$0, $QB_d^{fm}$1, and $QB_d^{fm}$2. In this normalization, the attracting fixed points are placed at $QB_d^{fm}$3 and $QB_d^{fm}$4, a repelling fixed point is placed at $QB_d^{fm}$5, and the remaining fixed points on the Julia set are ordered counterclockwise as $QB_d^{fm}$6 (He et al., 22 Jul 2025).

This marking convention canonically identifies the boundary dynamics with the monomial model. Given the model

$QB_d^{fm}$7

and a standard representative

$QB_d^{fm}$8

there is a unique homeomorphism $QB_d^{fm}$9 conjugating $B_d^{fm}$0 to $B_d^{fm}$1 and satisfying $B_d^{fm}$2. If

$B_d^{fm}$3

is an $B_d^{fm}$4-cycle of $B_d^{fm}$5 on $B_d^{fm}$6, then

$B_d^{fm}$7

is the multiplier of the image cycle $B_d^{fm}$8 for $B_d^{fm}$9. This defines holomorphic multiplier functions on $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$0, and for any $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$1, cycles on $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$2 are repelling, so $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$3.

The marked circle dynamics supply a concrete coordinate system for infinitesimal questions. The ordering of fixed points and the distinguished conjugacy $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$4 allow repelling multipliers on the Julia set to be tracked holomorphically across moduli, and those multiplier coordinates are later used in the analysis of pressure and Weil–Petersson degeneracy.

3. Complex structure and simultaneous uniformization

The complex structure on $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$5 is constructed indirectly from polynomial dynamics. Let $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$6 denote fixed-point-marked polynomials, and let $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$7 be the central hyperbolic component containing the marked monomial model. The central analytic statement is the biholomorphism

$B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$8

with the property that if $B_d^{fm}\times \overline{B_d^{fm}}\to QB_d^{fm}$9 and $\mathcal{SA}_d^{fm}$0 are the two Fatou components of $\mathcal{SA}_d^{fm}$1, then

$\mathcal{SA}_d^{fm}$2

Moreover, the restriction of $\mathcal{SA}_d^{fm}$3 to the diagonal $\mathcal{SA}_d^{fm}$4 is a diffeomorphism $\mathcal{SA}_d^{fm}$5. The holomorphic structure on $\mathcal{SA}_d^{fm}$6 is then defined by pulling back the holomorphic structure of $\mathcal{SA}_d^{fm}$7 via

$\mathcal{SA}_d^{fm}$8

(He et al., 22 Jul 2025).

In this framework, the main simultaneous uniformization theorem takes the form

$\mathcal{SA}_d^{fm}$9

a biholomorphism such that for fixed-point-marked Blaschke products $d\ge 2$0 and $d\ge 2$1, the map $d\ge 2$2 restricts on its two Fatou components to maps biholomorphically conjugate to $d\ge 2$3 and $d\ge 2$4, respectively. Here

$d\ge 2$5

The same paper shows that this map agrees, as a smooth map, with McMullen’s mating construction: $d\ge 2$6

The proof uses quasiconformal surgery and conformal welding on the unit circle. For standard representatives $d\ge 2$7, the mated Beltrami differential is defined by

$d\ge 2$8

and the measurable Riemann mapping theorem is used to solve for $d\ge 2$9. One then defines the resulting rational map piecewise on $d$0 and $d$1. Holomorphic dependence is established using holomorphic motions and analytic dependence of solutions of Beltrami equations. A distinctive technical point is that the quasi-Blaschke setting requires holomorphic dependence of sufficiently many boundary points, reflecting that

$d$2

which is larger than

$d$3

The formal analogy is with Bers’ simultaneous uniformization

$d$4

Here $d$5 plays the role of a “Teichmüller space” for the inside dynamics, and $d$6 plays the outside role.

4. Weil–Petersson and pressure semi-norms

For a smooth path $d$7 in $d$8 with standard representatives, one defines a holomorphic vector field $d$9 on $f:\mathbb P^1\to \mathbb P^1$0 by

$f:\mathbb P^1\to \mathbb P^1$1

where $f:\mathbb P^1\to \mathbb P^1$2 satisfy $f:\mathbb P^1\to \mathbb P^1$3 and extend quasiconformally to $f:\mathbb P^1\to \mathbb P^1$4. McMullen’s Weil–Petersson semi-norm is

$f:\mathbb P^1\to \mathbb P^1$5

and the paper proves the exact relation

$f:\mathbb P^1\to \mathbb P^1$6

so the Weil–Petersson and pressure semi-norms agree up to a factor of $f:\mathbb P^1\to \mathbb P^1$7 (He et al., 22 Jul 2025).

Degeneracy of the semi-norm has strong dynamical consequences. If

$f:\mathbb P^1\to \mathbb P^1$8

then

$f:\mathbb P^1\to \mathbb P^1$9

This connects metric degeneracy to infinitesimal multiplier rigidity. A second input is the holomorphic index formula: if $d+1$00 denotes the attracting multiplier, then for every $d+1$01,

$d+1$02

From this, degeneracy yields the identity

$d+1$03

where $d+1$04.

The super-attracting locus is

$d+1$05

and it is a codimension-$d+1$06 subspace of $d+1$07. The main non-degeneracy theorem states:

1. for any $d+1$08 and non-zero $d+1$09, one has $d+1$10;
2. for any $d+1$11 and $d+1$12, one has $d+1$13;
3. for Lebesgue almost every $d+1$14, one has $d+1$15.

The mechanism combines the holomorphic structure on $d+1$16, the identity

$d+1$17

and holomorphicity of the multiplier at the holomorphic attracting fixed point $d+1$18. In the paper’s formulation, Ivrii’s trick and a perpendicularity argument force a contradiction unless the holomorphic attracting multiplier vanishes. The metric therefore fails to degenerate away from $d+1$19, and even on $d+1$20 degeneracy is exceptional. A related path-length statement is that any non-trivial $d+1$21-path $d+1$22 has

$d+1$23

5. Low-degree cases and explicit computations

The general theory specializes cleanly in low degrees. For $d+1$24, the Weil–Petersson semi-norms are non-degenerate in $d+1$25; in this case $d+1$26 is a singleton, so non-degeneracy follows immediately from the general theorem. For $d+1$27, the Weil–Petersson semi-norms are likewise non-degenerate in $d+1$28 (He et al., 22 Jul 2025).

In degree $d+1$29, the super-attracting locus admits the explicit family

$d+1$30

The multiplier at the marked repelling fixed point $d+1$31 is

$d+1$32

Its derivatives satisfy

$d+1$33

These formulas show that the multiplier varies nontrivially along the family and rule out degeneracy of the Weil–Petersson semi-norm on $d+1$34.

These low-degree results are significant because they show that the semi-norm is not merely generically non-degenerate but fully non-degenerate in the first two nontrivial marked Blaschke moduli spaces. They also illustrate the role of marked multipliers as explicit diagnostic functions for infinitesimal geometry.

6. Position within the broader Blaschke-product literature

The fixed-point-marked quasi-Blaschke theory sits next to several adjacent Blaschke-product settings without collapsing into them. The paper on unicritical Blaschke products studies the normalized family

$d+1$35

and shows that the parabolic locus in parameter space is an epicycloid with $d+1$36 cusps,

$d+1$37

whose interior corresponds to elliptic maps with an attracting fixed point in $d+1$38; it does not define quasi-Blaschke products, but it gives a concrete parameter-space geometry for a restricted Blaschke family (Cao et al., 2015).

A different neighboring direction associates a finite Blaschke product

$d+1$39

to the local quasiregular model

$d+1$40

near a fixed point with constant complex dilatation. There the emphasis is on classifying fixed rays and switched antipodal pairs via boundary fixed points of $d+1$41; again, the paper does not use the term quasi-Blaschke product, but it gives a fixed-point-marked correspondence between local quasiregular dynamics and boundary dynamics of a finite Blaschke product (Fletcher, 2015).

Another distinct strand concerns maximal Blaschke products. There the central object is an extremal bounded analytic function with prescribed critical set, normalized at a fixed point, and the main result is that the extremal is an essentially unique indestructible Blaschke product. Fixed-point marking is implemented by conjugation with disk automorphisms, but the theory remains entirely conformal and does not define quasi-Blaschke products (Kraus et al., 2013).

Taken together, these neighboring theories clarify the specificity of the fixed-point-marked quasi-Blaschke moduli problem. The 2025 theory is not a reformulation of unicritical parameter spaces, local quasiregular models, or maximal-function extremal problems. Its characteristic features are the simultaneous uniformization

$d+1$42

the diagonal realization of the Blaschke locus, and the interaction between marked multiplier rigidity and the Weil–Petersson/pressure geometry. A plausible implication is that fixed-point marking provides the precise level of rigidification needed to transplant ideas from Bers-type uniformization and McMullen’s thermodynamic metric theory into the setting of rational dynamics with quasi-circular Julia sets.