Abstract

Three main results are presented in this thesis. The first is a proof of the existence of absolutely continuous invariant measures (ACIMs) for two dimensional maps supported on islands, which are small, disjoint regions of R2. The proof is computer-assisted and uses both numerical evidence and a combinatorial method. We give examples of weak chaos for which ACIMs exist: within islands there is chaos, but from a distance orbits are periodic.

The second main result is a geometrical proof of the asymptotic behavior of generalized tent maps with memory which we call elliptical maps. It is proved that for certain π-rational angles, all points in the domain except for (0, 0) fall into a polygonal region whose characteristics we determine. When the angles are π-irrational we prove that these points either fall in a unique ellipse, or accumulate on its boundary.

The third result is a proof that ACIMs exist for a certain range of parameters in generalized β-tent maps with memory.

The thesis begins with discussions about ACIMs and why they are interesting, followed by Tsujii’s theorem and other tools, notes on computer calculations and graphics, and on weak chaos. At the end, we highlight some unanswered questions and puzzling phenomena that we encountered during our research.