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We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ2↦ℂ2 under which all complete external rays from infinity for f have well defined endpoints.

:// [5] Field M.J., Dynamics and Symmetry, ICP Adv. Texts Math., 3, Imperial College Press, London, 2007 [6] Floer A., A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems, 1987, 7(1), 93–103 [7] Floer A., Zehnder E., The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ergodic Theory Dynam. Systems, 1988, 8, 87–97 [8] Gęba K., Degree for gradient

hyperbolas, 39 Hamiltonan-Jacobi equation, 56 Hamiltonian system, 11, 32, 36, 52, 55 Herman, 62 heteroclinic connection, 72 point, 72 point, transversal, 72 Hill’s regions, 127 hodograph, 40 homoclinic loop, 70, 74, 138, 178, 179, 182 loop, breaking, 72, 179 orbit, 138, 168, 169 orbit, transversal, 139, 169, 170 point, 70 point, transversal, 175 tangle, 168 tube, 139 tube, breaking, 139, 170, 178 hyperbolic saddle, 68 tangle, 97, 166, 168 hyperbolic invariant set, 71, 103, 104, 107, 109, 112, 168 hyperbolic orbits, 110 implicit function theorem, 80 impulsive maneuver, 117

capture transfer used by Hiten arriving at Moon on October 2, 1991. 147 3.16 Nominal exterior WSB-transfer. 150 3.17 Capture process. 154 3.18 Time sections ∑ t to periodic orbit ψ having a transverse homoclinic orbit φ. A map σ is defined on ∑ t with trans- verse homoclinic point r. The sections are all identified. 170 3.19 Intersection of Wu(0),Ws(0) on Q3 = 0. 173 3.20 Region near infinity in McGehee coordinates. 174 3.21 Map ψ near infinity. 175 3.22 Construction of the transversal map, φ̃. 177 3.23 Kepler flow in McGehee coordinates. 181 3.24 Hyperbolic invariant

Chapter Three Capture This chapter comprises about one-half of this book, and a number of different topics are covered pertaining to the subject of capture in the three- and four- body problems. Two forms of capture are studied, and in the last section we prove that they are equivalent on a hyperbolic invariant set for the circular restricted problem. Section 3.1 serves as an introduction to the different types of capture and other dynamics we will consider in this chapter. These include permanent capture and a theorem by Chazy and Hopf on the measure of

. Floer, A refinement of the Conley index and application to the stability of hyperbolic invariant sets, Erg. Th. and Dyn. Sys. 7 (1987), 93-103, [11] K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Birkhäuser, Topological Nonlinear Analysis, Degree, Singularity and Variations, Eds. M. Matzeu i A. Vignoli, Progr. in Nonl. Diff. Equat. and Their Appl. 27, Birkhäuser, (1997), 247-272, [12] A. Gołȩbiewska and S. Rybicki, Bifurcations of critical orbits of invariant strongly indefinite functionals, accepted for publication to Nonl. Anal. TMA

degenerate system (2.3) connecting normally hyperbolic invariant sets implies the existence of a transversal homoclinic orbit (transversal heteroclinic orbit) of system (2.1) for sufficiently small e. Now we consider the case of a non-transversal homoclinic orbit. To this end we assume (A3) For λ = λο, the degenerate system (2.3) has a hyperbolic equilibrium point u0 , that is, the Jacobian gu(uo,^o) has no eigenvalues on the imaginary axis. (A4) For λ = λ0 , the degenerate system (2.3) has a homoclinic orbit 7 to u0, that is, there is a solution 7 : R —> Rn of

chaotic process by proving the existence of a hyperbolic invariant set. There are several new theorems and results in this book that have not been previously published. These include Theorem 2.33 on the existence of Aubrey-Mather sets in the restricted problem, and Theorem 3.58 on the proof of existence of an invariant hyperbolic network on an extended version xvi PREFACE of the weak stability boundary. A short proof is given for Theorem 1.39 on the existence of geodesic flows for the Kepler problem. Scope of This Book The approach taken here is mathematical and is both

a specific set. 1.0.2 Limit of the first eigenvalue and the topological pressure Under hyperbolicity assumptions on the field b, the limit of the sequence " has been determined in [10, 31]. It was based on the fact that " is given by the following variational formula (see [10]), " D sup 2P.V / Z V cdC inf u>0 h Z V L".u/ u d i ; where belongs to P.V /, the space of probability measure on V . It is proved in [31] that when the recurrent set K of the field b is a finite union of isolated components K1; K2; : : : ; KN which are hyperbolic invariant sets, the limit

the transverse directions. A classical example of a strange attractor is the Lorenz attractor. There are many other examples of invariant sets with “wild” topological structure. Among them are the well-known Smale-Williams solenoid and Smale horseshoe. The latter is an example of a hyperbolic invariant set whose local topological structure is the product of different Cantor-like sets. Strange attractors are always associated with trajectories having extremely irregular behavior and are thought of as the origin of “dynamical” chaos. Specialists in dynamical