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We extend in several ways a recently proposed method to constructone-dimensional chaotic maps with exactly known natural invariant measure [Sogo1999, 2009]. First, we assume that the given invariant measure depends on acontinuous parameter and show how to construct maps with robust chaos --i.e.,chaos that is not destroyed by arbitrarily small changes of the parameter-- andprescribed invariant measure and constant Lyapunov exponent. Then, by relaxingone condition in the approach of Refs. \cite{Sogo1,Sogo2}, we describe a methodto construct robust chaos with prescribed constant invariant measure andvarying Lyapunov exponent. Another extension of a condition in Refs. [Sogo1999, 2009] provides a new method to get robust chaos with known varyingLyapunov exponent. In this third approach the invariant measure can be computedexactly in many particular cases. Finally we discuss how to use diffeomorphismsto construct maps with robust chaos, any number of parameters and prescribedinvariant measure and Lyapunov exponent.

Robust chaos with prescribed natural invariant measure and Lyapunov exponent