Renormalization in one-dimensional dynamics

The study of the dynamical and topological properties of interval exchange transformations and their natural generalizations is an important problem, which lies at the intersection of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory, and geometric group theory. The purpose of the survey is to make a systematic presentation of the existing results on the ergodic and geometric characteristics of the one-dimensional maps under consideration, as well as on the measured foliations on surfaces and two-dimensional complexes that one can associate with these maps. These results are based on the research of the ergodic properties of the renormalization process- an algorithm that takes an original dynamical system and builds a sequence of equivalent dynamical systems with a smaller support set. For all dynamical systems considered in the paper these renormalization algorithms can be viewed as multidimensional fraction algorithms.