Mappings with Three Homeomorphism Intervals that Change the Orientation of One of Them. A key result consists in the proof of existence of a wandering interval. Also, we prove that any mapping under consideration has exactly one invariant normalized measure. Ergodic Properties of the Horocycle Flow and Classification of Fuchsian Groups. We give several criteria for ergodicity and conservativity and connect them with the classification of the associated Fuchsian groups.
Special attention is given to covering surfaces.
In particular, we show that normal subgroups of divergent-type Fuchsian groups provide natural examples for the strictness of a number of inclusions in the classification of Fuchsian groups. On Orders of Movable Poles of the Schlesinger Equation. We treat here movable singular points of the Schlesinger equation of isomonodromic deformations as the points of jumping of the splitting type of the vector bundle constructed from the initial Fuchsian equation. We show that solutions to the Schlesinger equation have at these points poles of order two and we also illustrate the results mentioned above by means of some Fuchsian system of two differential equations with four singular points. Stochastic Stability in a Model of Drilling.