Let \(\{f_i\}_{i=1}^n\) be a finite collection of piecewise \(C^1\) maps from the unit interval \(X\) into itself and let \(\{p_i(x)\}_{i=1}^n\) be a probability distribution for each \(x\in X\). Then one defines a random map, i.e., a Markov chain in which at a point \(x\in X\) the map \(f_i\) is chosen with probability \(p_i(x)\). Following A. M. Vershik [Discrete Contin. Dyn. Syst. 13, No. 5, 1305–1324 (2005; Zbl 1115.37002)], the author calls this well-known object a “polymorphism”, adding however a property that it preserves the Lebesgue measure. If there are no other invariant measures absolutely continuous with respect to the Lebesgue measure, the author calls it ergodic. It is worth noting that this definition of ergodicity is rather nonstandard. For a very special random map composed of 2 piecewise linear elements and a constant probability distribution the author gives conditions under which the above mentioned ergodicity takes place.