In this paper we provide a connection between the geometrical properties of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.
Semiclassical scaling of eigenstate thermalization in single-particle chaotic systems
University of Science and Technology of China（中国科学技术大学）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Wilczek Quantum Center, Shanghai Institute for Advanced Studies（复旦张肇韦量子中心，上海高等研究院）；CAS Key Laboratory of Microscale Magnetic Resonance（中国科学院微观磁学共振重点实验室）
Temporal entanglement transition in chaotic quantum many-body dynamics
Princeton University（普林斯顿大学）；University of Geneva（日内瓦大学）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；University of California, Berkeley（加州大学伯克利分校）；National University of Singapore（新加坡国立大学）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）；KU Leuven（荷语鲁汶大学）；Google Research（谷歌研究）
