Necessary and sufficient conditions of full chaos for expanding Baker-like maps and their use in non-expanding Lorenz maps

In this work we give necessary and sufficient conditions for a discontinuous expanding map f of an interval into itself, made up of N pieces, to be chaotic in the whole interval. For N=2 we consider the class of expanding Lorenz maps, for N ≥ 3 a class of maps whose internal branches are onto, called Baker-like. We give the necessary and sufficient conditions for a discontinuous expanding map to be chaotic in the whole interval and persistent under parameter perturbations (robust full chaos in short). These classes of maps represent a suitable first return in non-expanding Lorenz maps. Thus the obtained conditions can be used to prove robust full chaos in non-expanding Lorenz maps. An example from the engineering application is illustrated.