In this talk we we will first examine the dynamical properties of the simplest form of a piecewise isometry in one dimension, the interval exchange tranformation. We will then generalize this concept to interval translation mappings, and examine their dynamical properties, and consider an example of a rank 3 ITM which is of infinite type.

Extended Harper's model arises in a tight binding description of 2
dimensional crystal layers subject to an external magnetic field. From a
dynamical point of view the model provides an example for a quasi-periodic
analytic Jacobi-cocyle with singularities.

In the first part of the talk, we show how to extend (and with what
limitations)
Avila's global theory of analytic SL(2,C) cocycles to families of cocycles
with singularities. In particular, we shall introduce a strategy of
computing the Lyapunov exponent valid for any analytic cocycle with
possible singularities.

As an application this allows to determine the Lyapunov exponent for
extended Harper's model, for all values of parameters, which so far did
not even exist on a heuristic level in physics literature.

In the second part of our talk, results on the spectral analysis of the
extended Harper's model will be presented.