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Hyperbolic Polygonal Billiards Close to 1-Dimensional Piecewise Expanding Maps

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Abstract

We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards have a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probabilities (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is set equal to zero. We prove that if a convex polygon satisfies a generic condition called (*), and the reflection law has a Lipschitz constant sufficiently small, then there exists a one-to-one correspondence between the ergodic SRB measures of the billiard map and the ergodic acips of the corresponding slap map, and moreover that the number of Bernoulli components of each ergodic SRB measure equals the number of the exact components of the corresponding ergodic acip. The case of billiards in regular polygons and triangles is studied in detail.

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Notes

  1. We might have as well chosen the extension to be right continuous.

  2. This notion is not exactly equal to the one given in [6]. According to this definition arbitrarily small perturbations of a polygon without parallel sides facing each other may have parallel sides facing each other.

  3. This choice is arbitrary, every \( {\bar{\alpha }}>2 \) will do.

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Acknowledgements

The authors were partially funded by the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017). The authors JLD and JPG were partially supported by the Project CEMAPRE-UID/MULTI/00491/2019 financed by FCT/MCTES through National Funds. The author PD was partially supported by the Strategic Project PEst-OE/MAT/UI0209/2013. All authors wish to thank the anonymous referees for reading carefully the manuscript and making many useful remarks and suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Gianluigi Del Magno

  2. Departamento de Matemática, CEMAPRE and REM, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781, Lisbon, Portugal

    João Lopes Dias & José Pedro Gaivão

  3. Departamento de Matemática and CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749-016, Lisbon, Portugal

    Pedro Duarte

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  1. Gianluigi Del Magno
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  2. João Lopes Dias
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  3. Pedro Duarte
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  4. José Pedro Gaivão
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Correspondence to Gianluigi Del Magno.

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Communicated by Eric A. Carlen.

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Del Magno, G., Lopes Dias, J., Duarte, P. et al. Hyperbolic Polygonal Billiards Close to 1-Dimensional Piecewise Expanding Maps. J Stat Phys 182, 11 (2021). https://doi.org/10.1007/s10955-020-02673-2

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