Study of large deviation principle for non-hyperbolic dynamical systems

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03359
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Nagaoka University of Technology
• Principal Investigator: Yamamoto Kenichiro 長岡技術科学大学, 工学研究科, 准教授 (30635181)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 大偏差原理 / 区分単調区間写像 / Markov Diagram

Outline of Final Research Achievements

In this research project, we consider a large deviation principle for non-hyperbolic dynamical systems. In particular, we prove that a transitive piecewise monotonic map with positive topological entropy satisfies a level-2 large deviation principle with a unique measure of maximal entropy as reference under the condition that the set of periodic measures is dense in the set of ergodic measures. This has applications to a broad class of piecewise monotonic maps including linear mod 1-transformations and piecewise monotonic maps with two monotonic pieces.

Free Research Field

エルゴード理論

Academic Significance and Societal Importance of the Research Achievements

力学系に関する大偏差原理の研究は約40年にわたり様々な研究者により継続的になされており、1990年頃にマルコフ型力学系に対して大偏差原理の成立が示されたが、非マルコフ型に関しては十分に研究が進んでいない状況である。本研究により今まで大偏差原理の成立が示されていなかった区分単調区間写像について大偏差原理の成立が示された。