Fixed-Point Marked Blaschke Products

• Fixed-point-marked Blaschke products are rational maps with explicit fixed-point data that facilitate unique moduli classifications and normalization.
• Their Möbius normalization yields canonical representations that expose boundary dynamics and underpin rigidity theorems.
• Applications include precise local quasiregular models, entropy formulas in random dynamics, and unified approaches to dynamical system classification.

Fixed-point-marked Blaschke products are Blaschke products studied together with explicit fixed-point data. In its most formal moduli-theoretic version, a degree-$d$ fixed-point-marked rational map is a pair $(f;x_1,\dots,x_{d+1})$ in which $f\in \mathrm{Rat}_d$ and the ordered tuple records all fixed points of $f$, counted with multiplicity; the corresponding moduli space of degree-$d$ fixed-point-marked Blaschke products is denoted $B_d^{fm}$ (He et al., 22 Jul 2025). Related work uses the same organizing idea more broadly: a distinguished interior fixed point, a boundary fixed point with angular derivative, or a local fixed-point germ may serve as the marking data that rigidifies a Blaschke product or makes it a canonical model for a wider dynamical system.

1. Definitions, normalizations, and marking data

A finite Blaschke product of degree $d\ge 2$ is a rational map

$$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$

where $\Delta$ is the unit disk. Such maps preserve $\Delta$, its exterior $(f;x_1,\dots,x_{d+1})$0, and the unit circle $(f;x_1,\dots,x_{d+1})$1, and their Julia set is exactly $(f;x_1,\dots,x_{d+1})$2 (He et al., 22 Jul 2025). In the fixed-point-marked setting, one passes from the unmarked map $(f;x_1,\dots,x_{d+1})$3 to the marked datum $(f;x_1,\dots,x_{d+1})$4, where the $(f;x_1,\dots,x_{d+1})$5 enumerate $(f;x_1,\dots,x_{d+1})$6.

The conjugation action of $(f;x_1,\dots,x_{d+1})$7 on fixed-point-marked rational maps is

$(f;x_1,\dots,x_{d+1})$8

Within $(f;x_1,\dots,x_{d+1})$9 and the broader quasi-Blaschke space $f\in \mathrm{Rat}_d$0, a standard representative is one for which $f\in \mathrm{Rat}_d$1 is a Blaschke product and

$f\in \mathrm{Rat}_d$2

For such a representative, the map has the explicit form

$f\in \mathrm{Rat}_d$3

with parameters $f\in \mathrm{Rat}_d$4 (He et al., 22 Jul 2025).

Outside the fully marked moduli-theoretic framework, normalization by automorphisms of the disk remains fundamental. For a holomorphic self-map $f\in \mathrm{Rat}_d$5, automorphisms

$f\in \mathrm{Rat}_d$6

move a distinguished point to $f\in \mathrm{Rat}_d$7, and the normalized maps

$f\in \mathrm{Rat}_d$8

fix $f\in \mathrm{Rat}_d$9. For finite Blaschke products, if $f$0, these normalized maps converge locally uniformly to the rotation

$f$1

which gives a boundary-asymptotic description of a moving marked point approaching $f$2 (Fricain et al., 2011).

2. Local fixed-point models and associated Blaschke products

A local fixed-point-marked model appears in the study of planar quasiregular maps with constant complex dilatation near a fixed point. The model map is

$f$3

with $f$4, $f$5, and $f$6. Its complex dilatation is constant,

$f$7

and the associated unicritical Blaschke product is

$f$8

The induced circle map $f$9 on directions satisfies $d$0, where $d$1 is the rescaling operator on degree-$d$2 circle maps (Fletcher, 2015).

In this setting, the marking data are precisely local fixed-point invariants. The degree $d$3 records the local index of the fixed point, and the parameter $d$4 records the local complex dilatation and stretching direction. Fixed rays of $d$5 correspond to fixed points of $d$6, and hence to fixed points of $d$7 on $d$8, with the known parity modification when $d$9 is odd. The paper defines $B_d^{fm}$0 to be elliptic, parabolic, or hyperbolic according to the Denjoy–Wolff type of the associated Blaschke product $B_d^{fm}$1, and this classification determines the number and type of invariant curves landing at the marked fixed point.

The local model becomes a conjugacy theorem for actual quasiregular germs. If $B_d^{fm}$2 is quasiregular near a fixed point $B_d^{fm}$3, has local index $B_d^{fm}$4, and has constant complex dilatation in a neighborhood of $B_d^{fm}$5, then there exist a neighborhood $B_d^{fm}$6, parameters $B_d^{fm}$7, and a quasiconformal map $B_d^{fm}$8 such that

$B_d^{fm}$9

on $d\ge 2$0, with $d\ge 2$1 asymptotically conformal at $d\ge 2$2. External rays $d\ge 2$3 then transfer the fixed-ray structure of $d\ge 2$4 to fixed or switched invariant curves of $d\ge 2$5 landing at $d\ge 2$6 (Fletcher, 2015). In this sense, the associated Blaschke product is a local fixed-point-marked model for the quasiregular germ.

3. Boundary fixed points, critical-set marking, and rigidity

A second major use of fixed-point marking is boundary rigidity. A Burns–Krantz type theorem states that if $d\ge 2$7 is holomorphic and $d\ge 2$8, then

$d\ge 2$9

forces $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$0 to be the identity. In the Blaschke setting, if $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$1 is a maximal Blaschke product for $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$2, $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$3 is a boundary fixed point of $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$4 with finite positive angular derivative, and there is a non-tangential sequence $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$5 such that

$$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$6

then $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$7 on all of $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$8 (Moucha, 27 May 2025). Here the fixed-point marking is the boundary germ at $$f(z)=e^{2\pi i\theta}\prod_{i=1}^{d}\frac{z-a_i}{1-\overline{a_i}z}, \qquad (a_1,\dots,a_d)\in \Delta^d,\ \theta\in \mathbb{R}/\mathbb{Z},$$9, while the maximal Blaschke product condition encodes the critical-set marking.

Maximal Blaschke products arise from an extremal problem with a distinguished basepoint. Given an $\Delta$0-critical set $\Delta$1, let $\Delta$2 be the multiplicity of $\Delta$3 in $\Delta$4. Kraus and Roth consider

$\Delta$5

where $\Delta$6 consists of bounded analytic functions whose critical set contains $\Delta$7. They prove that the extremal function $\Delta$8 is unique, is an indestructible Blaschke product, has critical set exactly $\Delta$9, is normalized by $\Delta$0 and $\Delta$1, and is finite exactly when $\Delta$2 is finite (Kraus et al., 2013). This is a canonical origin-marked construction; by disk automorphisms it becomes a fixed-point-marked construction at any interior point.

Prescribing fixed points on the boundary can also be done explicitly. Given pairwise distinct $\Delta$3, there exists at least one finite Blaschke product $\Delta$4 with

$\Delta$5

All such solutions of degree at most $\Delta$6 are parameterized by admissible tuples $\Delta$7 through an explicit formula, and the associated boundary multipliers satisfy

$\Delta$8

In this fully fixed boundary case, the identity map is the only solution of degree $\Delta$9 (Bolotnikov, 2016). This yields a concrete boundary fixed-point-marked family in which the marked points are $(f;x_1,\dots,x_{d+1})$00 and the angular derivatives are part of the parameter data.

4. Unicritical families, elliptic loci, and fixed-point geometry

For unicritical finite Blaschke products of degree $(f;x_1,\dots,x_{d+1})$01, there is a normal form

$(f;x_1,\dots,x_{d+1})$02

and every unicritical Blaschke product of degree $(f;x_1,\dots,x_{d+1})$03 is Möbius-conjugate to a unique such $(f;x_1,\dots,x_{d+1})$04 (Fletcher, 2014). The parameter $(f;x_1,\dots,x_{d+1})$05 marks the unique critical point in $(f;x_1,\dots,x_{d+1})$06, while the Denjoy–Wolff point provides the relevant fixed-point datum.

The elliptic locus

$(f;x_1,\dots,x_{d+1})$07

consists of parameters for which the Denjoy–Wolff point lies in $(f;x_1,\dots,x_{d+1})$08. Its rotationally symmetrized version $(f;x_1,\dots,x_{d+1})$09 is a starlike domain about $(f;x_1,\dots,x_{d+1})$10 containing the disk

$(f;x_1,\dots,x_{d+1})$11

and the corresponding connectedness locus $(f;x_1,\dots,x_{d+1})$12 is $(f;x_1,\dots,x_{d+1})$13 together with finitely many parabolic boundary points (Fletcher, 2014). The fixed-point meaning is direct: inside $(f;x_1,\dots,x_{d+1})$14, the marked attracting fixed point lies in the disk; on the boundary, it becomes parabolic on $(f;x_1,\dots,x_{d+1})$15; outside, the Denjoy–Wolff point is a boundary attracting fixed point.

For the degree-$(f;x_1,\dots,x_{d+1})$16 unicritical family, the relative boundary of the full elliptic locus is the epicycloid

$(f;x_1,\dots,x_{d+1})$17

These are exactly the parabolic parameters, and when $(f;x_1,\dots,x_{d+1})$18 the Denjoy–Wolff point is

$(f;x_1,\dots,x_{d+1})$19

Thus the boundary of the elliptic region is parameterized by the parabolic fixed point itself (Cao et al., 2015). In degree $(f;x_1,\dots,x_{d+1})$20, this curve is a cardioid, and every degree-$(f;x_1,\dots,x_{d+1})$21 Blaschke product is unicritical in the disk; the paper further constructs an explicit conjugacy invariant $(f;x_1,\dots,x_{d+1})$22 so that a degree-$(f;x_1,\dots,x_{d+1})$23 product is elliptic, parabolic, or hyperbolic according as $(f;x_1,\dots,x_{d+1})$24 lies inside, on, or outside the cardioid (Cao et al., 2015).

5. The moduli space $(f;x_1,\dots,x_{d+1})$25: complex structure and pressure geometry

The most explicit theory of fixed-point-marked Blaschke products is the complex-analytic study of $(f;x_1,\dots,x_{d+1})$26. Let $(f;x_1,\dots,x_{d+1})$27 be the connected component of fixed-point-marked quasi-Blaschke products containing the model

$(f;x_1,\dots,x_{d+1})$28

and let $(f;x_1,\dots,x_{d+1})$29. Then $(f;x_1,\dots,x_{d+1})$30 is the locus represented by actual Blaschke products (He et al., 22 Jul 2025).

A simultaneous uniformization theorem identifies this space with a diagonal in a product of polynomial hyperbolic components. There is a biholomorphism

$(f;x_1,\dots,x_{d+1})$31

such that the restriction of $(f;x_1,\dots,x_{d+1})$32 to the diagonal

$(f;x_1,\dots,x_{d+1})$33

is a diffeomorphism onto $(f;x_1,\dots,x_{d+1})$34. Equivalently, there is a biholomorphism

$(f;x_1,\dots,x_{d+1})$35

realizing fixed-point-marked quasi-Blaschke products as matings of two fixed-point-marked Blaschke products. The complex structure on $(f;x_1,\dots,x_{d+1})$36 is defined by pulling back the complex structure of $(f;x_1,\dots,x_{d+1})$37 through this diagonal identification (He et al., 22 Jul 2025).

This moduli space carries dynamical multiplier functions. For each repelling cycle $(f;x_1,\dots,x_{d+1})$38 of the model map $(f;x_1,\dots,x_{d+1})$39, there is a corresponding holomorphic multiplier function $(f;x_1,\dots,x_{d+1})$40 on $(f;x_1,\dots,x_{d+1})$41. Every Blaschke product also has a holomorphic attracting fixed point in $(f;x_1,\dots,x_{d+1})$42 with multiplier $(f;x_1,\dots,x_{d+1})$43, and the super-attracting locus

$(f;x_1,\dots,x_{d+1})$44

is a complex codimension-$(f;x_1,\dots,x_{d+1})$45 subspace (He et al., 22 Jul 2025).

The same paper studies the pressure/Weil–Petersson geometry of $(f;x_1,\dots,x_{d+1})$46. The Weil–Petersson semi-norm is non-degenerate outside $(f;x_1,\dots,x_{d+1})$47; if $(f;x_1,\dots,x_{d+1})$48, it is still non-degenerate on directions transverse to $(f;x_1,\dots,x_{d+1})$49, and for Lebesgue almost every tangent vector inside $(f;x_1,\dots,x_{d+1})$50. In degrees $(f;x_1,\dots,x_{d+1})$51 and $(f;x_1,\dots,x_{d+1})$52, the Weil–Petersson semi-norm is everywhere non-degenerate on $(f;x_1,\dots,x_{d+1})$53 and $(f;x_1,\dots,x_{d+1})$54, hence defines a genuine metric there (He et al., 22 Jul 2025). Fixed-point marking is essential in this theory because it removes quotient singularities coming from permuting fixed points and makes the multiplier coordinates globally meaningful.

6. Extensions, variants, and broader uses of fixed-point marking

In random dynamics, the marking may itself vary measurably. For an admissible random Blaschke product cocycle $(f;x_1,\dots,x_{d+1})$55, there exists a measurable map

$(f;x_1,\dots,x_{d+1})$56

such that

$(f;x_1,\dots,x_{d+1})$57

for $(f;x_1,\dots,x_{d+1})$58-almost every $(f;x_1,\dots,x_{d+1})$59. The random invariant measure has density

$(f;x_1,\dots,x_{d+1})$60

on $(f;x_1,\dots,x_{d+1})$61, so the invariant measure is canonically centered at the random fixed point $(f;x_1,\dots,x_{d+1})$62. The fibre entropy is then

$(f;x_1,\dots,x_{d+1})$63

and averaging over rotations yields an entropy formula independent of the particular random marking (González-Tokman et al., 15 May 2025).

A different variant appears in finite-dimensional families of Blaschke-type rational maps modeling multimodal circle maps. For each integer $(f;x_1,\dots,x_{d+1})$64, the family

$(f;x_1,\dots,x_{d+1})$65

realizes all post-critically finite $(f;x_1,\dots,x_{d+1})$66-multimodal circle maps satisfying the paper’s dynamical hypotheses, and the realization is unique up to rotation (Faria et al., 7 May 2026). A plausible implication is that adding a fixed-point normalization would remove this residual rotation ambiguity and convert the “unique up to rotation” statement into uniqueness in a marked moduli space.

More generally, boundary normalization remains a recurrent theme. When a marked point approaches the unit circle, the normalized Blaschke products

$(f;x_1,\dots,x_{d+1})$67

converge to a rotation determined by the boundary derivative, which can be read as the asymptotic local model at a moving marked fixed point on $(f;x_1,\dots,x_{d+1})$68 (Fricain et al., 2011). Across local quasiregular dynamics, maximal Blaschke products, boundary rigidity, unicritical parameter spaces, and moduli theory, the fixed-point-marked viewpoint serves as a unifying device: it replaces coarse conjugacy classes by objects with enough normalization data to support explicit formulas, rigidity theorems, and analytic structures.