In this thesis, we study integrable birational systems, which are discrete integrable systems represented by the dynamics of birational maps. Integrability of such systems can be characterized by the degree growth of the birational maps, which is also closely related to the singularity structures of the systems. Therefore, the problem of computing the degree growth is of vital importance in terms of understanding the integrability of birational systems. We devote one part of the thesis to the problem of degree growth, with the focus on studying the degree growth of integrable birational systems by understanding the singularity confinement patterns of the systems. The remaining parts of the thesis are devoted to geometric constructions of integrable birational systems in dimensions N=2 and N=3. By geometric construction we mean that we construct integrable maps using some prescribed geometric data which are preserved by our construction. For example, in dimension N=2, we construct integrable maps by composing involutions defined on pencils of elliptic curves. These maps preserve a pencil of elliptic curves by construction and are therefore integrable. We generalize the constructions to dimension N=3 by considering involutions which are defined on pencils of quadrics. The birational maps constructed in this way can be considered as 3-dimensional generalizations of QRT maps, which coincide with QRT maps on the projective plane P¹¹ when restricted to any nondegenerate quadric. One can further generalize the construction by considering pertubations to 3D QRT maps which preserves the singularity confinement patterns. It turns out that the maps obtained using this method are actually autonomous versions of discrete Painlev\'e equations, which are usually considered as two-dimensional non-autonomous systems. Following this idea, we are able to construct 3D autonomous discrete Painlev\'e equations from pencils of quadrics.
Kangning Wei (Thu,) studied this question.