We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.

The invariant Cantor sets of the logistic map gμ(x) = μx(1 - x) for μ > 4 are hyperbolic and form a continuous family. We show that this family can be obtained explicitly through solutions of initial value problems for a system of infinitely coupled differential equations due to the hyperbolicity. The same result also applies to the tent map Ta(x) = a(1/2 - |1/2 - x|) for a > 2.

Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height E c there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > E c there are no bounded orbits. They called the bifurcation at E = E c an abrupt bifurcation to chaotic scattering.The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the antiintegrable limit, and to do so for a general range of chaotic scattering problems.

Chaos Hausdorff dimensionIn this paper, we prove that the β-transformations are chaotic in the sense of both LiYorke and Devaney. The topological and metric properties of the sets of points with dense or non-dense orbits are investigated. We also prove the result that the set of points with non-dense orbits under the β-transformation is of full Hausdorff dimension for any β > 1.

The orbital fates in a planar three-center problem of slightly negative energy are studied numerically, by classifying the dynamics into colliding with one of the centers, escaping to infinity and orbiting around the centers. We consider three energy levels, for each of them, the set of initial values leading to collision with the centers is determined. It presents an intriguing fractal structure. The complementary set, which corresponds to those initial values whose orbits are either regular or chaotic winding around the fixed centers, also exhibits a fractal structure. These fractal structures found here might lead to some observable physical feature in the future. The fractal structures imply the sensitivity to initial conditions, thus enable us to find initial values with which the orbits are chaotic, having a positive Lyapunov exponent.