Rational functions sharing preimages and height functions
Fedor Pakovich
TL;DR
Let $A,B$ be non-constant rational functions and $K$ an infinite set contained in $\mathbb{P}^1(\boldsymbol{k})$ with $\boldsymbol{k}$ finitely generated, or a discrete infinite subset when $A,B$ are polynomials. The paper proves that if $A^{-1}(K)\subseteq B^{-1}(K)$, then ${\rm deg}\,B\ge{\rm deg}\,A$, and hence $A^{-1}(K)=B^{-1}(K)$ is impossible unless ${\rm deg}\,A={\rm deg}\,B$; it also derives corollaries about infinite completely invariant sets of the correspondence $(\mathbb{P}^1,A,B)$. The main method combines height machinery, notably the Moriwaki height, with orbit considerations of the multivalued map $H(z)=B(A^{-1}(z))$, augmented by a simple height argument for the polynomial/discrete case. These results generalize prior compact-$K$ findings to broad infinite sets, revealing rigidity phenomena in the dynamics of rational maps and providing tools for studying invariant sets of rational correspondences.
Abstract
Let $A$ and $B$ be non-constant rational functions over $\mathbb{C}$, and let $K \subset \mathbb{P}^1(\mathbb{C})$ be an infinite set. Using height functions, we prove that the inclusion $ A^{-1}(K) \subseteq B^{-1}(K) $ implies the inequality $ {\rm deg} B \geq {\rm deg} A $ in the following two cases: the set $K$ is contained in $\mathbb{P}^1(k)$, where $ k$ is a finitely generated subfield of $\mathbb{C}$, or the set $K$ is discrete in $\mathbb{C}$, and $A$ and $B$ are polynomials. In particular, this implies that for $A$, $B$, and $K$ as above, the equality $ A^{-1}(K) = B^{-1}(K) $ is impossible, unless $ {\rm deg} B = {\rm deg} A $.