Ulam's method for computing stationary densities of invariant measures for piecewise convex maps with countably infinite number of branches

Let τ: I=[0, 1]→ [0, 1] be a piecewise convex map with countably infinite number of branches. In {GIR}, the existence of absolutely continuous invariant measure (ACIM) μ for τ and the exactness of the system (τ, μ) has been proven. In this paper, we develop an Ulam method for approximation of f^*, the density of ACIM μ. We construct a sequence \τₙ\ₙ₌₁^∞ of maps τₙ: I→ I s. t. τₙ has a finite number of branches and the sequence τₙ converges to τ almost uniformly. Using supremum norms and Lasota-Yorke type inequalities, we prove the existence of ACIMs μₙ for τₙ with the densities fₙ. For a fixed n, we apply Ulam's method with k subintervals to τₙ and compute approximations fn,k of fₙ. We prove that fn,k→ f^* as n→ ∞, k→ ∞, both a.e. and in L¹. We provide examples of piecewise convex maps τ with countably infinite number of branches, their approximations τₙ with finite number of branches and for increasing values of parameter k show the errors \|f^*-fn,k\|₁.