Statistical properties of type D dispersing billiards

Margaret Brown; Péter Nándori

doi:10.3934/dcds.2022073

Article Contents

2022, Volume 42, Issue 10: 4823-4851. Doi: 10.3934/dcds.2022073

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Statistical properties of type D dispersing billiards

Margaret Brown^1, and
Péter Nándori^2, ,

1.
Penn State University, State College, PA, 16802, USA
2.
Yeshiva University, New York, NY, 10016, USA

^*Corresponding author: Péter Nándori
^*Corresponding author: Péter Nándori

Received: August 2020

Revised: December 2021

Early access: June 2022

Published: September 2022

MB was partially supported by NSF DMS 1800811 and PN was partially supported by NSF DMS 1800811 and NSF DMS 1952876

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Abstract

We consider dispersing billiard tables whose boundary is piecewise smooth and the free flight function is unbounded. We also assume there are no cusps. Such billiard tables are called type D in the monograph of Chernov and Markarian [9]. For a class of non-degenerate type D dispersing billiards, we prove exponential decay of correlation and several other statistical properties.

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Mathematics Subject Classification: Primary: 37C83; Secondary: 37A50.

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Figure 1. A scatterer with a convex corner point (left) and a concave corner point (right)

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Figure 2. A simple corridor bounded by two corner points

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Figure 3. Singular trajectories after a long flight. The trajectory on the top panel is tangent to the scatterer on the left and the trajectory on the bottom panel is tangent to the scatterer on the right. A neighborhood of the first collision point is magnified for better visibility

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Figure 4. Singularity structure near type 3 and type 1 boundary points Similar figures can be found in [4,Figure 11]. An unstable curve is indicated with bold on both panels

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