Smooth dependence of fixed points of Hamiltonians. (English) Zbl 1500.37041
Summary: 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.
MSC:
| 37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
| 37J38 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 15A23 | Factorization of matrices |
Keywords:
symplectic Hamiltonian; positive definite matrix; Thompson metric; discrete algebraic Riccati equation; Stein operatorReferences:
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