Diophantine Properties of Dynamical Systems: Quantitative, Connectivity and Proximality

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The central topic of this project is Diophantine and Combinatorial Properties of Dynamical Systems in general and Interval Exchange Transformations (IETs) in particular. In 1993 the PI, Boshernitzan, initiated study of Quantitative Recurrence (QR) obtaining a quantitative enhancement of the celebrated Poincare recurrence theorem claiming that typical orbits of dynamical systems (under some mild conditions) return arbitrarily close to their initial position. The quantitative enhancement addresses the question how soon and how close this recurrence takes place. It turns out that the answer to these questions is possible in terms of the Hausdorff dimension of the space. In the last two years the PI and his student, Jon Chaika, initiated research that studyies in a quantitative manner the Connectivity and Proximality phenomena in dynaThe Connectivity property of a dynamical system manifests itself in that, for a typical pair of points in the space, the closures of the orbits of these points coincide. Connectivity is more restrictive than Recurrence. Certain assumptions (like minimality or ergodicity) are imposed for the phenomenon to hold. The Proximality property manifests itself in that a typical pair of points is proximal, i.e. their orbits approach each other arbitrary close. The phenomenon of Proximality is not universal (but it holds for weakly-mixing systems). Some initial results on Quantitative Connectivity and Proximality (QCP) were obtained recently in joint work by the PI and his student, Chaika, where a number of important related questions were posed. These results generalize and strengthen some classical results in Diophantine analysis (in Number theory) and are related to the recent work of others in the area of "shrinking targets" and "monotone shrinking target property" of dynamical systems. The results are detailed and complete and of interest when applied to IETs. IETs are invertible piecewise orientation preserving isometries of the unit interval into itself and their study is motivated not only by their intrinsic interest as generalizations of the rotations of the unit circle and the simplest invertible maps which preserve Lebesgue measure, but also by their applications such as to area preserving flows in compact surfaces, and in Teichmuller theory. The study of QR initiated in 1993 attracted a lot of attention in the national and international mathematical community motivating much further research and development in this and related areas. The QCP project is a natural continuation of the QR project and it is hoped that it will have an impact on research in Dynamical Systems and related fields similar that of the CR. Many open problems and various connections with other fields of mathematics (Ergodic Theory, IETs, Teichmuller Theory, area preserving flows in compact surfaces), may keep researchers in the subject busy for years to come. Throughout the last 28 years, PI Boshernitzan has been responsible for the selection and coaching the Rice team (and all interested students) for the Putnam mathematical competition. He has mentored graduate students, the most recent one is Jon Chaika (2010) who himself has earned much recognition by the mathematical community.