On a Very Steep Version of the Standard Map

Parameter α ∈ R is called the twist parameter. A point at position (x, y) on the cylinder moves at constant height y around the cylinder a distance αy, and then moves up one unit if it is on the right half of the cylinder (x′ ∈ (1/2, 1)), down one unit if it is on the left half (x′ ∈ (0, 1/2)), and stays at the same vertical position if it is at the singular lines x′ = 1/2 or 0. Such system can be regarded as a discontinuous analog of the standard Chirikov map (see [3]), where the smooth function sin(x′) is replaced by the discontinuous sgn(x′). This system can be also obtained from the Fermi-Ulam accelerator model with the sawtoothlike wall movement regime (see [1] for details). For the smooth variants of the described problems KAMtechnique can be used to provide the existence of the