Smooth dependence of fixed points of Hamiltonians

We consider the Lie semigroup of symplectic Hamiltonians acting on the open convex cone of positive definite matrices via linear fractional transformations. Each member of its interior contracts strictly the invariant Finsler metric, the Thompson metric on the cone, and has a unique positive definite fixed point. We show that the fixed point map is smooth. As applications, we obtain the smooth dependence of the solutions of discrete algebraic Riccati equations and a family of smooth maps from the Siegel upper half-plane over the cone of positive definite matrices into its imaginary part. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this study, we investigate the Lie semigroup of symplectic Hamiltonians acting on positive definite matrices via linear fractional transformations. Our findings include the strict contraction of the invariant Finsler metric and the Thompson metric for each member of the interior, as well as the existence of a unique positive definite fixed point.