...Figure 10.4: Invariant manifold M near the origin. This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both...

... the critical energy. The proof of Theorem 4.5 consists of three steps. The first proves a Kupka-Smale-like theorem about non-degeneracy of periodic orbits. The second shows that a non-degenerate locally minimal orbit is always hyperbolic. The third finishes the proof by proving the finite local families...

... theory and its relevance to molecular dynamics. The chapter establishes the connection with kinetic theory, followed by an exploration of unstable periodic orbits and the Ruelle zeta function, highlighting their significance in understanding molecular behaviour. The implications of these probabilistic...

...This chapter studies complex polynomials with only one critical point, relating arithmetic properties of the coefficients to those of periodic orbits and their multipliers and external rays. It first defines the complex polynomial maps of degree n ≥ 2, and draws an alternate normal...