On the ergodic properties of piecewise linear perturbations of the twist map

Abstract

Dedicated to the memory of V. M. AlexeyevAbstract. It is proved that for a sequence of arbitrarily small piecewise linear perturbations of the twist map, there is a domain with stochastic behaviour (almost hyperbolicity). The measure of this domain has the asymptotics uA In-^-(1+0 (1) ), A->0A where A is the magnitude of the perturbation.

Invariant circles for the piecewise linear standard map

“…We also compute "windows" in the parameter range where there are no invariant circles whatever. Examples of invariant circles conjugate to rational rotations were first found by Wojtkowski [12,13], who also carried out a detailed investigation of the mixing properties of h k .…”

A system of one dimensional balls with gravity

“…Nonetheless their actual or possible presence thwarts attempts to study the mixing component since the two components have to be intricately intertwined. So far only crude models were found where the two types of behavior were shown to coexist ( [Wl,Prl,W2,Del]) but the success there depends on a simple splitting of the phase space which is destroyed under any kind of perturbation. Thus for the case of mixed behavior we have virtually no examples and no theorems about the mixing component.…”

A criterion for the positivity of the Liapunov characteristic exponent

“…[3], [4]). The case C = 2 corresponds to perturbations of the twist map and was treated in [4] and [5]. For C = -2, -1 , 1, one can get results similar to those of [4] by essentially the same approach (this was done explicitly in [1]).…”

“…The criterion is an abstraction of methods used in proving positivity of the m.L.c.e. for some piecewise linear tranformations of the torus [4], [1], [5]. In § 3, we give another application of the criterion in the same spirit.…”