Abstract
As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.
Highlights
An exponential dichotomy (ED for short) is a hyperbolic splitting of the extended state space for linear nonautonomous differential or difference equations into two bundles of linear subspaces: The so-called stable vector bundle consists of all solutions decaying exponentially in forward time, while the complementary unstable vector bundle consists of all solutions which exist and decay in backward time
EDs turned out to be an ambient hyperbolicity concept when dealing with nonautonomous dynamical systems: In stability theory the associated dichotomy spectrum of a linearization yields the appropriate uniform asymptotic stability, while gaps in this spectrum give rise to invariant manifolds or fiber bundles
In nonautonomous bifurcation theory it plays a crucial role, how invariant projectors or the associated invariant vector bundles for EDs behave under parameter-variation
Summary
[BV11] recently proved that dichotomy projectors vary continuously differentiable, provided a dichotomic difference equation is subject to small perturbations w.r.t. the C1-norm. We demonstrate that the smooth dependence of dichotomy projectors on the difference equation is merely a corollary from established proof techniques (cf., for example, [San93, AM96]).
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