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Journal of the Australian Mathematical Society, vol.119, no.1, pp.106-127, 2025 (SCI-Expanded)
In the early 1900s, Maillet [Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions (Gauthier-Villars, Paris, 1906)] 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 dimensions turns out to fail. In fact, using a result by Kleinbock and Margulis ['Flows on homogeneous spaces and Diophantine approximation on manifolds', Ann. of Math. (2) 148(1) (1998), 339-360], we show that among analytic matrix functions in dimension n\ge 2$ ]]>, Maillet's invariance property is only true for Möbius transformations with special coefficients. This implies that the analogue in higher dimensions of an open question of Mahler on the existence of transcendental entire functions with Maillet's property has a negative answer. However, extending a topological argument of Erdos ['Representations of real numbers as sums and products of Liouville numbers', Michigan Math. J. 9 (1962), 59-60], 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 Alniaçik and Saias ['Une remarque sur les G-denses', Arch. Math. (Basel) 62(5) (1994), 425-426], and show that the situation depends on the matrix dimensions m,n$ ]]>. Finally, we discuss extensions of a related result by Burger ['Diophantine inequalities and irrationality measures for certain transcendental numbers', Indian J. Pure Appl. Math. 32 (2001), 1591-1599] to quadratic matrices. We state several open problems along the way.