Maillet's property and Mahler's Conjecture on Liouville numbers fail for matrices

Abstract

In the early 1900's, Maillet proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimension turns out to fail. In fact, using a result by Kleinbock and Margulis, we show that among analytic matrix functions in dimension $n\ge 2$, Maillet's invariance property is only true for M\"obius transformations with special coefficients. This implies that the analogue in higher dimension of an open question of Mahler on the existence of transcendental entire functions with Maillet's property has a negative answer. On the other hand, extending a topological argument of Erd\H{o}s, we prove that for any injective continuous self mapping on the space of rectangular matrices, many Liouville matrices are mapped to Liouville matrices. Dropping injectivity, we consider setups similar to Alnia\c{c}ik and Saias and show that the situation depends on the matrix dimensions $m,n$. Finally we discuss extensions of a related result by Burger to quadratic matrices. We state several open problems along the way.