Volume 78, Nº 6 (2023)

The study of dynamical and topological properties of interval exchange transformations and their natural generalisations is an important problem that is placed at the crossroad of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory and geometric group theory. The current survey is focused on the systematic presentation of the results about ergodic and geometric properties of the orbits of certain one-dimensional maps. We also apply it to get information on the properties of the leaves of associated measured foliations on surfaces and 2-dimensional complexes . These results are based on the study of the properties of the renormalization processes. In each case the renormalisation can be seen as an algorithm that starts with a dynamical system and builds up an equivalent system with a smaller support set. For each class of dynamical systems that we deal with (interval exchange transformations with flips, linear involutions, interval translation maps, systems of isometries) we provide a brief description of the corresponding renormalization process. We show which properties of interval exchange transformations can be generalised and which ones change crucially. We also formulate the most challenging open problems. In the last section we provide a detailed description of the topological interpretation of the obtained results, including several applications to the Novikov's problem of asymptotic behaviour of plane sections of triply periodic surfaces.