In this paper we prove the existence of full measure unbounded chaotic attractors which are persistent under parameter perturbation (also called robust). We show that this occurs in a discontinuous piecewise smooth one dimensional map f, belonging to the family known as Nordmark’s map. To prove the result we extend the properties of a full shift on a finite or infinite number of symbols to a map, here called Baker like map with infinitely many branches, defined as a map of the interval I=[0,1] in to itself with infinitely branches due to expanding functions with range I except at most the rightmost one. The proposed example is studied by using the first return map in I, which we prove to be chaotic in I making use of the border collision bifurcations curves of basic cycles. This leads to a robust unbounded chaotic attractor, the interval (−∞,1], for the map f.
