15 research outputs found
Unique ergodicity of circle and interval exchange transformations with flips
We study the existence of transitive exchange maps with flips defined on the
unit circle. We provide a complete answer to the question of whether there
exists a transitive exchange map of the unit circle defined on n subintervals
and having f flips.Comment: 13 pages, 6 figures; notational changes, smaller figure
On two-dimensional surface attractors and repellers on 3-manifolds
We show that if is an -diffeomorphism with a surface
two-dimensional attractor or repeller and is a
supporting surface for , then and
there is such that: 1) is a union
of disjoint tame surfaces such that every is
homeomorphic to the 2-torus . 2) the restriction of to
is conjugate to Anosov automorphism of
Introduction to the qualitative theory of dynamical systems on surfaces
This book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achievements in this area obtained in recent times by Russian and foreign mathematicians whose work has not yet appeared in the monographic literature. The main stress here is put on global problems in the qualitative theory of flows on surfaces. Despite the fact that flows on surfaces have the same local structure as flows on the plane, they have many global properties intrinsic to multidimensional systems. This is connected mainly with the existence of nontrivial recurrent trajectories for such flows. The investigation of dynamical systems on surfaces is therefore a natural stage in the transition to multidimensional dynamical systems. The reader of this book need be familiar only with basic courses in differential equations and smooth manifolds. All the main definitions and concepts required for understanding the contents are given in the text. The results expounded can be used for investigating mathematical models of mechanical, physical, and other systems (billiards in polygons, the dynamics of a spinning top with nonholonomic constraints, the structure of liquid crystals, etc.). In our opinion the book should be useful not only to mathematicians in all areas, but also to specialists with a mathematical background who are studying dynamical processes: mechanical engineers, physicists, biologists, and so on