Existence of ACIM for Piecewise Expanding $C^{1+\varepsilon}$ maps

In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron Operator of a piecewise expanding $C^{1+\varepsilon}$ map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem \cite{ionescu1950}, 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.