Over the past almost three decades dynamical systems have played a central role in spectral analysis of quasiperiodic Hamiltonians as well as certain quasiperiodic models in statistical mechanics (most notably: the Ising model, both quantum and classical). There are many ways of introducing quasiperiodicity into a model. We shall concentrate on the widely studied Fibonacci case (which is a prototypical example of so-called substitution systems on two letters with certain desirable properties). In this case a particular geometric scheme, arising from a certain smooth three-dimensional dynamical system associated to the quasiperiodic model in question (the so-called Fibonacci trace map) has been established. Our aim is to present a general dynamical/geometric framework and to demonstrate how information about the model in question (spectral properties for Hamiltonians, and Lee-Yang zeros distribution for classical Ising models) can be obtained from the aforementioned dynamical system and the geometry of certain dynamically invariant sets. In this first in a series of two (or three) talks, we'll briefly recall how dynamical systems are associated to Schroedinger and Jacobi operators, as well as classical Ising models. We'll establish notation, ask main questions and in general prepare the ground for a somewhat more general (in terms of geometry and dynamical systems) discussion for next time.