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Universal measure for Poncelet-type theorems


Authors: Evgeny A. Avksentyev and Vladimir Yu. Protasov
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 53A04, 28A25, 51N15
DOI: https://doi.org/10.1090/proc/13838
Published electronically: August 10, 2018
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Abstract: We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measure are well known and have been used in the literature to prove Poncelet's and the zigzag theorems. Some further generalizations are also obtained by applying the new measure.


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Additional Information

Evgeny A. Avksentyev
Affiliation: Department of Mechanics and Mathematics of Moscow State University, Moscow, 111991 Russia
Email: avksentjev@mail.ru

Vladimir Yu. Protasov
Affiliation: DISIM of the University of L’Aquila, 67100 L’Aquila, Italy and Department of Mechanics and Mathematics of Moscow State University, Moscow, 111991 Russia
Email: v-protassov@yandex.ru

DOI: https://doi.org/10.1090/proc/13838
Keywords: Emch theorem, Poncelet theorem, zigzag theorem, invariant measure, quadric, pencil of circles
Received by editor(s): June 10, 2015
Received by editor(s) in revised form: March 22, 2017
Published electronically: August 10, 2018
Additional Notes: The second author was supported by RFBR grants no. 16-04-00832 and 17-01-00809, and by RSCF grant no. 17-11-01027
Communicated by: Michael Wolf
Article copyright: © Copyright 2018 American Mathematical Society

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