  In this paper, we prove the quasi-compactness of the Frobenius-Perron
operator for a piecewise convex map $\tau$ with a countably infinite number of
branches on the interval $I=[0,1]$. We establish that for high enough $n$
iterates of $\tau$, $\tau^n$ are piecewise expanding. Using the Lasota-Yorke
Inequality derived from references \cite{hofbauer1982} and \cite{keller1985},
adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic
theorem, we demonstrate the existence of absolutely continuous invariant
measure (ACIM) $\mu$ for $\tau$, the exactness of the dynamical system $(I,
\tau,\mu)$ and the quasi-compactness of Frobenius-Perron operator $P_\tau$
induced by $\tau$. The last fact implies a multitude of strong ergodic
properties of $\tau$.
