Thermodynamic formalism for conformal semigroup actions

2016 Fiscal Year Final Research Report

• Project/Area Number: 15H06416
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Shimane University
• Principal Investigator: Jaerisch Johannes 島根大学, 総合理工学研究科, 講師 (90741869)
• Research Collaborator: SUMI Hiroki 京都大学, 大学院人間・環境学研究科共生人間学専攻数理科学講座, 教授 (40313324) ; MATSUZAKI Katsuhiko 早稲田大学, 教育・総合科学学術院, 教授 (80222298) ; KESSEBÖHMER Marc University Bremen ; STADLBAUER Manuel University of Rio de Janeiro ; MUNDAY Sara University of Bologna
• Project Period (FY): 2015-08-28 – 2017-03-31
• Keywords: Ergodic theory / Dynamical systems / Fractal Geometry / Hyperbolic Geometry / Geometric group theory

Outline of Final Research Achievements

We have investigated the interplay of dynamics and geometry of conformal semigroup actions. Two of the main results are the following.
1) We have applied the multifractal analysis to random complex dynamical systems. In particular, we have investigated the Hoelder exponent of the long-term behavior depending on the initial value (J. Jaerisch, H. Sumi, Adv. Math 313 (2017), 36 pages).
2) We have investigated the Poincare exponent of abstract free groups and its normal subgroups. In the proof we use the discrete Laplacian on the associated Cayley graph (J. Jaerisch, K. Matsuzaki, Proc. AMS, to appear 2017).

Free Research Field

Ergodic theory and dynamical systems