Let x and y be points in a billiard table M = M (σ) in R 2 that is bounded by a curve σ. We assume σ ∈ Σr with r ≥ 2, where Σr is the set of simple closed C r curves in R 2 with positive curvature. A subset B of M \ {x, y} is called a blocking set for the pair (x, y) if every billiard path in M from x to y passes through a point in B. If a finite blocking set exists, the pair (x, y) is called secure in M ; if not, it is called insecure. We show that for σ in a dense G δ subset of Σr with the C r topology, there exists a dense G δ subset R = R(σ) of M (σ) × M (σ) such that (x, y) is insecure in M (σ) for each (x, y) ∈ R. In this sense, for the generic Birkhoff billiard, the generic pair of interior points is insecure. This is related to a theorem of S. Tabachnikov [24] that (x, y) is insecure for all sufficiently close points x and y on a strictly convex arc on the boundary of a smooth table.

Let x and y be points in a billiard table M = M (σ) in R 2 that is bounded by a curve σ. We assume σ ∈ Σ r with r ≥ 2, where Σ r is the set of simple closed C r curves in R 2 with positive curvature. A subset B of M \ {x, y} is called a blocking set for the pair (x, y) if every billiard path in M from x to y passes through a point in B. If a finite blocking set exists, the pair (x, y) is called secure in M ; if not, it is called insecure. We show that for σ in a dense G δ subset of Σ r with the C r topology, there exists a denseIn this sense, for the generic Birkhoff billiard, the generic pair of interior points is insecure. This is related to a theorem of S. Tabachnikov [20] that (x, y) is insecure for all sufficiently close points x and y on a strictly convex arc on the boundary of a smooth table.

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We consider the long time behavior of the trajectories of the discontinuous analog of the standard Chirikov map. We prove that for some values of parameters all the trajectories remains bounded for all time. For other set of parameters we provide an estimate for the escape rate for the trajectories and present a numerically supported conjecture for the actual escape rate.