Mathematical  Culture and Thought

Mathematical Culture and Thought

Against Cavalieri’s Principle? A Paradoxical Example in Dynamics

Document Type : Survey

Authors
1 1,3School of Mathematics, IPM, Iran
2 2Department of Mathematical Sciences, Sharif University of Technology, Iran
3 School of Mathematics, IPM, Iran
Abstract
There exists a continuous family of disjoint smooth curves, partitioning
the unit square, such that one can choose only one point from each curve to construct a
measurable set whose area is equal to 1. This contradicts our intuition which resembles
Cavalieri’s principle and Fubini’s theorem! So, what is the problem? In this note, we
discuss the theorem (due to A. Katok) and answer this question.
Keywords

Subjects


[1] Anosov, D. V., Ergodic properties of geodesic flows on closed Riemannian manifolds of negative
curvature , Soviet. Math. Dokl., 4, 1153–1156; English translation from Dokl. Akad. Nauk, 151
(1963), 1250-1252.
[2] Anosov, D. V., Sinaĭ, Ya. G., Some smooth ergodic systems, With an appendix by G. A. Margulis
(in Russian), Uspekhi Mat. Nauk 22, 107–172; translation from Uspekhi Mat. Nauk, 22 (1967),
107-172.
[3] Avila, A., Viana, M., Wilkinson, A., Absolute continuity, Lyapunov exponents and rigidity, I”
Geodesic flows, J. Eur. Math. Soc. (JEMS) 17 (2015) No. 6, 1435-1462.
[4] Brin, M., Stuck, G., Introduction to Dynamical Systems, Cambridge University Press, Cambridge,
2015.
[5] Camacho, C., Lins Neto, A., Geometric Theory of Foliations (transl. from the Portuguese by Sue E.
Goodman), Birkhäuser, Boston-Base-Stuttgart, 1985.
[6] Gogolev, A., How typical are pathological foliations in partially hyperbolic dynamics: an example,
Israel J. Math., 187 (2012), 493-507.
[7] Hirayama, M., Pesin, Y., Non-absolutely continuous foliations, Israel J. Math., 160 (2007), 173-
187.
[8] Hasselblatt, A., Anatole Katok—A half-century of dynamics, Notices Amer. Math. Soc., 66 (2019),
708-719.
[9] Hirsch, M. W., Pugh, C. C., Shub, M., Invariant Manifolds, Lecture Notes in Mathematics, vol. 583,
Springer-Verlag, Berlin-Heidelberg-New York, 1977.
[10] Kleiner, I.,  History of the infinitely small and the infinitely large in calculus, transl. R. Jahanipour, S. Maghsoudi,  Mathematical Culture and Thought, 65 (2019), 77-121. [in Persian]
[11] Lam, L.-Y., Shen, K., The Chinese concept of Cavalieri’s principle and its applications, Historia
Math., 12 (1985), 219-228.
[12] Milnor, J., Fubini foiled: Katok’s paradoxical example in measure theory, Math. Intelligencer, 19
(1997), 30-32.
[13] Milnor, J.,  Fubini foiled: Katok’s paradoxical example in measure theory, Nashr-e Riyazi, 18 (1998), 24-26. [in Persian]
[14] O’Connor, J. J., Robertson, E. F., MacTutor History of Mathematics
Archives, “Overview of Chinese mathematics” (February 2000), available at
https://mathshistory.st-andrews.ac.uk/HistTopics/Chinese_overview.
[15] Ruelle, D., Wilkinson, A., Absolutely singular dynamical foliations, Commun. Math. Phys., 219
(2001), 481-487.
[16] Shen, K., Crossley, J. N., Lun, A. W-C., The Nine Chapters on the Mathematical Art, Companion
and Commentary, Oxford University Press, Beijing, 1999.
[17] Shub, M., Wilkinson, A., Pathological foliations and removable zero exponents, Invent. Math., 139
(2000), 495-508.
[18] Stein, E. M., Shakarchi, R., Real Analysis: Measure Theory, Integration, and Hilbert Spaces,
Princeton University Press, NJ, 2005.
[19] Stillwell, J. C., Archimedes’ Lost Method (14 December 2007), in Encyclopedia Britannica, available
at https://www.britannica.com/topic/Archimedes-Lost-Method-1084593.
[20] Struik, D. J., A Concise History of Mathematics, Dover Publications, New York, 1987.
[21] Viana, M., Yang, J., Physical measures and absolute continuity for one-dimensional center direction,
Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 30 (2013), 845-877.

  • Receive Date 22 June 2021
  • Revise Date 16 November 2021
  • Accept Date 08 May 2022
  • Publish Date 21 January 2023