Determine the invariant density for the Example 2 piecewise convex map

Determine the explicit stationary density f* (with respect to the Lebesgue measure) of the absolutely continuous invariant measure for the interval map τ:[0,1]→[0,1] defined by τ(x) = 1/(((2i+1)/(i(i+1))) − x) − i on each subinterval [1/(i+1), 1/i] for integers i ≥ 1. The map τ has countably many branches and belongs to the class T_pc^{∞,0}(I), so an absolutely continuous invariant measure exists; the task is to derive the closed-form expression for its invariant density f*.

Background

The paper develops Ulam’s method to approximate stationary densities of absolutely continuous invariant measures (ACIMs) for piecewise convex maps with countably infinite branches. Existence and uniqueness of ACIMs are established for the classes T_pc^∞(I) and T_pc^{∞,0}(I).

In Section 6, Example 2 introduces a specific piecewise convex map τ on [0,1] with countably many branches, given by explicit formulas on intervals [1/(i+1), 1/i]. It is known, via prior results and Theorem 3 in the paper, that τ has an ACIM. The authors state that the exact invariant density for this map is not known, motivating the use of Ulam approximations. This makes the derivation of a closed-form invariant density for τ a concrete unresolved problem.