Une nouvelle preuve du théorème de point fixe de Handel
M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 348 (1996) 2637--2662] S Matsumoto [Topol. Appl. 104 (2000) 191--214]) to study homeomorphisms of surfaces.
Invariance of complementary domains of a fixed point set
It is an elementary exercise to verify that g is continuous. By a well-known theorem of M. H. A. Newman [3] (cf. also [1, p. 157]) g cannot be the identity on a nonempty open set. Hence U and f(U) are the only components of M F and f(f(U)) = U, thus proving the Proposition. In the case of (2) the above argument shows that F cannot contain an open set, hence dim F < (dim M) -1, and since F separates M we have dim F = (dim M) -1. In a refinement of Smith Theory, G. Bredon [2] has shown that if M is also orientable then any involution with an odd codimensional fixed point set must reverse the orientation; hence we obtain
LYAPUNOV-BASED TRANSFER BETWEEN ELLIPTIC KEPLERIAN ORBITS
We present a study of the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory specific to this problem. The construction of Lyapunov functions is based on the fact that a non-degenerate Keplerian orbit is uniquely described by its angular momentum and Laplace (- Runge-Lenz) vectors. We suggest a Lyapunov function, which gives a feedback controller such that the target elliptic orbit becomes a locally asymptotically stable periodic orbit in the closed-loop dynamics. We show how to perform a global transfer between two arbitrary elliptic orbits based on the local transfer result. Finally, a second Lyapunov function is presented that works only for circular target orbits.