Deck Groups in Bicritical Dynamics

• Deck groups are sets of Möbius transformations that leave a rational map invariant by satisfying f ∘ μ = f, defining the symmetry of its dynamics.
• In bicritical rational maps, the structure of deck groups is determined by degree parity, with non-power maps maintaining fixed cyclic or dihedral orders and power maps exhibiting unbounded cyclic growth.
• The classification uses dynamical and group-theoretic methods to show that iterative symmetry changes occur only under strict conditions related to critical point preservation.

In complex dynamics, the deck group of a rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ is the group of Möbius transformations $\mu$ satisfying $f\circ\mu=f$. For a bicritical rational map—one with exactly two critical points—the paper "On the deck groups of iterates of bicritical rational maps" gives a complete description, up to isomorphism, of the groups that occur as $\Deck(f^k)$ for iterates $f^k$ (Koch et al., 2022). The classification separates sharply by the parity of the degree and by whether the map is a power map, and it shows that the symmetries of iterates are highly constrained.

1. Definitions and basic setting

A bicritical rational map is a rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ with exactly two critical points, that is, $|\mathcal{C}_f|=2$ (Koch et al., 2022). A bicritical map is a power map if its set of critical points coincides with its set of critical values, $\mathcal{C}_f=\mathcal{V}_f$. Such maps are conjugate to $z\mapsto z^{\pm d}$ for $d\geq 2$ (Koch et al., 2022).

For any rational map $\mu$0, the deck group is

$\mu$1

where $\mu$2 is the group of Möbius transformations (Koch et al., 2022). The object of study is not only $\mu$3 itself, but the full sequence of deck groups of iterates,

$\mu$4

The central problem is to determine which abstract groups arise as $\mu$5 when $\mu$6 ranges over bicritical rational maps and $\mu$7 is arbitrary. The paper resolves this completely (Koch et al., 2022).

2. Classification by degree parity

The classification depends first on whether the degree $\mu$8 of $\mu$9 is odd or even (Koch et al., 2022).

For odd degree, the situation is rigid. If $f\circ\mu=f$0 is a bicritical map of odd degree $f\circ\mu=f$1, then $f\circ\mu=f$2 for all $f\circ\mu=f$3 if and only if $f\circ\mu=f$4 is not a power map. Moreover, $f\circ\mu=f$5 is a power map if and only if there exists $f\circ\mu=f$6 such that $f\circ\mu=f$7 for some $f\circ\mu=f$8 (Koch et al., 2022). Thus, in odd degree, non-power maps exhibit no growth in deck-group type under iteration, whereas power maps are exactly those for which larger cyclic groups appear.

For even degree, the range of possibilities expands, but only modestly. If $f\circ\mu=f$9 is a bicritical map of even degree $\Deck(f^k)$0, then for each $\Deck(f^k)$1, $\Deck(f^k)$2 is isomorphic to either $\Deck(f^k)$3, $\Deck(f^k)$4, or $\Deck(f^k)$5 for some $\Deck(f^k)$6 (Koch et al., 2022). If $\Deck(f^k)$7 is not a power map, then $\Deck(f^k)$8 for all $\Deck(f^k)$9 (Koch et al., 2022). This bound is sharp: for even degrees, both $f^k$0 and $f^k$1 are realized as deck groups of some iterate of some bicritical map (Koch et al., 2022).

The resulting classification can be summarized as follows.

Degree of $f^k$2	If $f^k$3 is a power map	If $f^k$4 is not a power map
Odd ($f^k$5)	$f^k$6	$f^k$7 for all $f^k$8
Even ($f^k$9)	$f:\widehat{\mathbb C}\to\widehat{\mathbb C}$0	$f:\widehat{\mathbb C}\to\widehat{\mathbb C}$1, $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$2, or $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$3
$f:\widehat{\mathbb C}\to\widehat{\mathbb C}$4	$f:\widehat{\mathbb C}\to\widehat{\mathbb C}$5	$f:\widehat{\mathbb C}\to\widehat{\mathbb C}$6

This suggests that iteration creates new symmetries only in tightly controlled circumstances, with the power-map case providing the only unbounded cyclic growth (Koch et al., 2022).

3. The quadratic and low-order cases

The quadratic case $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$7 is singled out explicitly in the classification (Koch et al., 2022). Here the possible deck groups are

$f:\widehat{\mathbb C}\to\widehat{\mathbb C}$8

If $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$9 is not a power map, then $|\mathcal{C}_f|=2$0 for all $|\mathcal{C}_f|=2$1 (Koch et al., 2022).

This case illustrates the even-degree pattern in its smallest instance. The cyclic possibilities $|\mathcal{C}_f|=2$2 correspond to the power-map behavior, while the non-power possibilities are uniformly bounded and reduce to the Klein four-group $|\mathcal{C}_f|=2$3 or the dihedral group $|\mathcal{C}_f|=2$4 (Koch et al., 2022). A plausible implication is that quadratic bicritical dynamics already displays the full structural dichotomy of the general theory: power maps allow iterative growth, whereas non-power maps allow only bounded symmetry enlargement.

4. Structural constraints on deck groups

A key structural theorem is the fixed-critical-point property: for a bicritical rational map $|\mathcal{C}_f|=2$5 and any $|\mathcal{C}_f|=2$6 for any $|\mathcal{C}_f|=2$7,

$|\mathcal{C}_f|=2$8

Thus every deck transformation of every iterate permutes the critical points of the original map (Koch et al., 2022). This is the basic mechanism restricting the allowable symmetry groups.

The group-theoretic reduction uses the classification of finite subgroups of Möbius transformations: they are cyclic, dihedral, or polyhedral ($|\mathcal{C}_f|=2$9, $\mathcal{C}_f=\mathcal{V}_f$0, $\mathcal{C}_f=\mathcal{V}_f$1). For bicritical maps and their iterates, however, only cyclic and dihedral groups arise (Koch et al., 2022). In odd degree, dihedral and polyhedral groups cannot occur, because these would require existence of involutions, which would force the degree to be even (Koch et al., 2022).

Another constraint comes from the geometry of the critical and critical-value sets. The set of Möbius transformations simultaneously preserving the 2-point sets of critical values and critical points is very limited, with at most four such transformations possible, namely the Klein four-group $\mathcal{C}_f=\mathcal{V}_f$2, and most often only the identity if $\mathcal{C}_f=\mathcal{V}_f$3 (Koch et al., 2022). This sharply limits the amount of symmetry that can persist across iterates.

5. Iteration, stabilization, and growth of $\mathcal{C}_f=\mathcal{V}_f$4

The sequence $\mathcal{C}_f=\mathcal{V}_f$5 does not grow arbitrarily. The paper shows that the deck group of an iterate strictly contains that of the previous iterate only under very restrictive circumstances tied to the action on critical values (Koch et al., 2022). In particular, if $\mathcal{C}_f=\mathcal{V}_f$6 ever stabilizes in the sense that

$\mathcal{C}_f=\mathcal{V}_f$7

then it remains constant for all further iterates (Koch et al., 2022).

For non-power bicritical maps, this stabilization phenomenon combines with the parity-based classification to give especially strong rigidity. In odd degree, the deck groups remain $\mathcal{C}_f=\mathcal{V}_f$8 for all $\mathcal{C}_f=\mathcal{V}_f$9; in even degree, they remain within the bounded list $z\mapsto z^{\pm d}$0, $z\mapsto z^{\pm d}$1, or $z\mapsto z^{\pm d}$2, with order at most $z\mapsto z^{\pm d}$3 (Koch et al., 2022).

By contrast, for power maps the explicit model $z\mapsto z^{\pm d}$4 satisfies

$z\mapsto z^{\pm d}$5

for all $z\mapsto z^{\pm d}$6 (Koch et al., 2022). This is the canonical example of iterative growth. The classification therefore separates bicritical maps into two dynamical symmetry regimes: bounded symmetry for non-power maps, and exponentially growing cyclic symmetry for power maps.

6. Examples, methods, and consequences

The paper gives explicit examples realizing the extremal even-degree possibilities. For

$z\mapsto z^{\pm d}$7

one has

$z\mapsto z^{\pm d}$8

and for

$z\mapsto z^{\pm d}$9

one has

$d\geq 2$0

(Koch et al., 2022). These constructions show that the upper bounds in the even-degree classification are not merely formal.

The proofs combine dynamical and group-theoretic methods. Dynamically, the action of Möbius transformations on critical points, critical values, and iterate structure controls the possible symmetries. Group-theoretically, the argument uses properties of finite subgroups of $d\geq 2$1. Commutative diagrams are used heavily to analyze when deck groups can grow in size with increasing $d\geq 2$2 (Koch et al., 2022).

The broader consequence is a precise classification of automorphism types possible in the iterated dynamics of bicritical maps (Koch et al., 2022). The only possible symmetries of iterates are cyclic or, when the degree is even, small dihedral extensions; in the generic non-power-map case, the deck group does not grow with iteration and symmetries are highly constrained (Koch et al., 2022). This suggests a strong rigidity principle for bicritical dynamics on the Riemann sphere: the presence of additional iterative symmetry is exceptional and is essentially characterized by the power-map condition.