Combinatorial Ergodic theory

摘要：DMS-0244994：宾州州立大学Kra提出了遍历理论和微分动力学的几个领域的研究，重点是遍历理论在数论和组合学中的应用。 主要的研究领域是“非常规”遍历平均的研究，研究遍历定理的平均功能，即使系统是遍历的，限制不需要是常数。 研究者提出了学习多项式平均值、交换变换的平均值和沿其他递归序列的平均值。 其他的研究是基于她在圆映射上的工作，包括映射的可微性，其旋转数的算术性质及其不变极限集的维数之间的联系。该项目研究其动力学过于复杂或过于混乱而无法局部理解的系统的长期行为。 动力系统被用来模拟自然科学和社会科学中的问题，对它们的研究有助于理解模拟现象的演变。 该项目的教育部分旨在为宾夕法尼亚州立大学的研究生和本科生课程做出贡献，特别课程旨在将研究注入早期阶段的学习过程。2003年秋季，PI将在MASS（数学高级研究学期）计划中开发和教授一门关于拓扑动力学和拉姆齐理论的课程，该计划是一个长期计划，旨在培养在国家竞争中脱颖而出的四年级本科生。 外联部分旨在使宾夕法尼亚州立大学数学系成为一个对妇女更有吸引力的学习和工作场所，建立一个妇女问题焦点小组，并支持女本科生的研究项目。

Abstract: DMS-0244994: Byrna KraPenn State UniversityKra proposes research in several areas of ergodic theory anddifferentiable dynamics, with an emphasis on applications of ergodictheory to number theory and combinatorics. The major area of research isthe study of ``non-conventional'' ergodic averages, studying ergodictheorems for averages of functions where even if the system is ergodic,the limit need not be constant. The investigator proposes studyingpolynomial averages, averages for commuting transformations and averagesalong other sequences of recurrence. Additional lines of research arebased on her work on circle maps, including connections between thedifferentiability of the map, the arithmetic properties of its rotationnumber and the dimension of its invariant limit set.The project studies the long term behavior of systems whose dynamics istoo complicated or too chaotic to be understood locally. Dynamicalsystems are used to model problems in natural and social sciences andtheir study leads to an understanding of the evolution of the modeledphenomena. The educational portion of the project aims to contribute tothe graduate and undergraduate programs at Penn State University, withspecial courses designed to inject research into the learning process atan early stage. In fall 2003, the PI will develop and teach a course ontopological dynamics and Ramsey Theory in the MASS (Mathematics AdvancedStudies Semesters) program, a long-standing program aimed at talented fourth-year undergraduates chosen in national competition. The outreach portion aims to make theMathematics Department at Penn State a more attractive place for women tostudy and work, with the creation of a focus group on women's issues andsupport of research projects for undergraduate women.