The aim of this work is to study a certain class of spatial polygons and prove theorems on the minimal number of flattenings that such polygons must have. In order to do this, we investigate spherical polygons which are not contained in any closed hemisphere and deduce, among many results, that under certain hypotheses such spherical polygons have a nontrivial lower bound on the number of spherical inflections.

The study of statistical properties of dynamical systems has been an active research area in recent decades.  Its main goal is to investigate when certain deterministic chaotic systems exhibit stochastic behavior when examined through the lens of a relevant invariant measure. Some of the key tools employed in deriving such results  are the spectral properties of the transfer operator. However, certain skew product systems, including random and mixed random-quasiperiodic linear cocycles, do not fit this approach. Very recent works have obtained limit laws  for these systems by studying the Markov Operator. The purpose of this dissertation  is to explain how these operators can be used to derive limit laws, such as Large Deviations Estimates and Central Limit Theorem, for certain skew-product dynamical systems.

We consider skew products over Bernoulli shifts, whose fibred dynamics is given by diffeomorphisms of the interval. We study the statistical and/or non-statistical behavior, referring to convergence and/or non-convergence, almost everywhere, of the Birkhoff average, respectively. We employ the Schwarzian derivative of the fiber maps and the arc-sine law to identify conditions under which these skew products exhibit these types of behavior. We identify distinct types of behavior according to the Schwarzian derivative. When the Schwarzian derivative is negative, the skew product has intermingled basins. Conversely, when the Schwarzian derivative is positive, the skew product has a physical measure. Finally, when the Schwarzian derivative is zero, the skew product has non-statistical behavior. In the latter scenario, we establish a connection between non-statistical behavior and the arc-sine law that allows us to obtain results in other settings independent of the sign of the Schwarzian derivative.

Algorithms relating Jacobi matrices and spectral variables are standard objects in numerical analysis. The recent discovery of bidiagonal coordinates led to the search of an appropriate algorithm for these new variables. The new algorithm is presented and compared with previous techniques.