Abstract

The main objective of this PhD thesis is to prove the quasi-compactness of the
Frobenius-Perron operator for two distinct types of interval maps. In the first part, we focus on
piecewise convex maps with an infinite number of branches defined on the unit interval [0, 1].
We show that for sufficiently high iterates, these maps exhibit piecewise expanding behavior. We
prove the Lasota-Yorke inequality for the pair (BV, L^1) and using the Ionescu Tulcea-Marinescu
theorem, we establish the existence of an absolutely continuous invariant measure (ACIM) and
the quasi-compactness of the Frobenius-Perron operator associated with these maps, revealing a
range of strong ergodic properties for the system. Additionally, we demonstrate the exactness of
the dynamical system.
In the second part of this thesis, we investigate piecewise expanding C^(1+ε) interval maps.
By proving a Lasota-Yorke inequality for the Frobenius-Perron operator for appropriate spaces
and again using the Ionescu Tulcea- Marinescu theorem, we show the existence of an ACIM.
Furthermore, we establish the quasi-compactness of the Frobenius-Perron operator for these maps
and explore key dynamical properties of the system, such as weak mixing and exponential decay
of correlations.