Quasi-compactness of Frobenius–Perron operator for piecewise expanding C1+𝜀 maps of an interval

In this paper, we establish Lasota–Yorke inequality for the Frobenius–Perron Operator of a piecewise expanding [Formula: see text] map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we prove the quasi-compactness of the Frobenius–Perron operator induced by the map. Additionally, we explore significant properties of the system, including weak mixing and exponential decay of correlations.