tailieunhanh - OF INVARIANT FIBER BUNDLES FOR DYNAMIC EQUATIONS ON MEASURE CHAINS ¨ CHRISTIAN POTZSCHE AND STEFAN

OF INVARIANT FIBER BUNDLES FOR DYNAMIC EQUATIONS ON MEASURE CHAINS ¨ CHRISTIAN POTZSCHE AND STEFAN SIEGMUND Received 8 August 2003 m -SMOOTHNESS We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here, an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the “Hadamard-Perron theorem” to the time-dependent, infinitedimensional, noninvertible, and parameter-dependent case, where the linear part is not necessarily hyperbolic with variable growth rates. As a key feature, our proof works without using complicated technical tools. 1. Introduction. | m-SMOOTHNESS OF INVARIANT FIBER BUNDLES FOR DYNAMIC EQUATIONS ON MEASURE CHAINS CHRISTIAN POTZSCHE AND STEFAN SIEGMUND Received 8 August 2003 We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the Hadamard-Perron theorem to the time-dependent infinitedimensional noninvertible and parameter-dependent case where the linear part is not necessarily hyperbolic with variable growth rates. As a key feature our proof works without using complicated technical tools. 1. Introduction The method of invariant manifolds was originally developed by Lyapunov Hadamard and Perron for time-independent diffeomorphisms and ordinary differential equations at a hyperbolic fixed point. It was then extended from hyperbolic to nonhyperbolic systems from time-independent and finite-dimensional to time-dependent and infinitedimensional equations and turned out to be one of the main tools in the contemporary theory of dynamical systems. It is our objective to unify the difference and ordinary differential equations case and extend them to dynamic equations on measure chains or time scales closed subsets of the real line . Such equations additionally allow to describe for example a hybrid behavior with discrete and continuous dynamical features or allow an elegant formulation of analytical discretization theory if variable step sizes are present. This paper can be seen as an immediate continuation of 18 where the existence and 1-smoothness of invariant fiber bundles for a general class of nonautonomous nonin-vertible and pseudohyperbolic dynamic equations on measure chains have been proved moreover we obtained a higher-order smoothness for invariant fiber bundles of stable and unstable types therein. While the existence and 1-smoothness result in 18 is a special .