Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${\displaystyle {\left(X_i\right)}^{\infty }\_\{i=1\}}$ and a sequence of continuous maps ${\displaystyle {\left(f_i\right)}^{\infty }\_\{i=1\}}$, ${\displaystyle f_i:X_i\to X_{i+1}}$, is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of ${\displaystyle f_n○...○f_2○f_1}$. As an application we construct a large class of smooth triangular maps of the square of type ${\displaystyle 2^{\infty }}$ and positive topological entropy.