Central Limit theorems and Packing Problems in Dynamics and Geometry

The present thesis describes two applications of Busemann-like functions to problems in dynamics and geometry. Busemann-like functions are used in various fields of mathematics to describe the distance to a point at infinity, and generalize the concept of horofunction. They describe the large-scale geometry of geodesics in a Hadamard spaces, and have recently been successfully used to show probabilistic properties. The first application uses Busemann-like functions to derive a Central Limit Theorem (CLT) for matrix valued cocycles where the process is driven by a topological Markov chain. We state a CLT in the non-commutative setting of random matrix products where the underlying process is driven by a subshift of finite type (SFT) with Markov measure. For the proof we use the martingale method introduced by Y. Benoist and J.F. Quint in the iid setting. Busemann-like cocycles via an auxiliary boundary space let us extend a centered cocycle and apply Brown's Martingale CLT to obtain the result. The second application is a direct use of the Busemann function to a packing problems in hyperbolic geometry. We give new packing density lower bounds for horoball packings in hyperbolic 5-space $\mathbb{H}^5$, realized in eleven different configurations within Coxeter simplex tilings. They are described in terms of the Busemann function, a hyperbolic isometry invariant, and appear in a commensurability class of arithmetic Coxeter groups. The Busemann functions paramerize the type of the various horospheres relative to a marked point (origin) $o \in \mathbb{H}^5$. Varying this parameter allows continuous transitions between the extremal optimal packing configurations. The new packing density lower bound is given in terms of the Riemann zeta function to be $\frac{5}{7 \zeta(3)}$. We conjecture this is optimal.