[{"oa":1,"scopus_import":"1","oa_version":"Published Version","OA_place":"publisher","year":"2025","has_accepted_license":"1","page":"467-503","acknowledgement":"I am very grateful to Vadim Kaloshin for suggesting the topic, his guidance during this project, and many helpful comments on an earlier version of the manuscript. Moreover, I would like to thank Comlan Edmond Koudjinan and Volodymyr Riabov for interesting discussions. Partial financial support by the ERC Advanced Grant ‘RMTBeyond’ No. 101020331 is gratefully acknowledged. This project received funding from the European Research Council (ERC) ERC Grant No. 885707.","license":"https://creativecommons.org/licenses/by/4.0/","department":[{"_id":"LaEr"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_created":"2024-09-22T22:01:43Z","file_date_updated":"2025-01-13T08:51:40Z","abstract":[{"lang":"eng","text":"It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus."}],"file":[{"checksum":"650fe115d998fe0ac3a8d0c7519447c8","access_level":"open_access","date_created":"2025-01-13T08:51:40Z","file_size":659100,"relation":"main_file","date_updated":"2025-01-13T08:51:40Z","file_name":"2025_ErgodicTheory_Henheik.pdf","content_type":"application/pdf","creator":"dernst","file_id":"18828","success":1}],"article_type":"original","title":"Deformational rigidity of integrable metrics on the torus","author":[{"id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","last_name":"Henheik","full_name":"Henheik, Sven Joscha"}],"citation":{"apa":"Henheik, S. J. (2025). Deformational rigidity of integrable metrics on the torus. <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/etds.2024.48\">https://doi.org/10.1017/etds.2024.48</a>","ieee":"S. J. Henheik, “Deformational rigidity of integrable metrics on the torus,” <i>Ergodic Theory and Dynamical Systems</i>, vol. 45, no. 2. Cambridge University Press, pp. 467–503, 2025.","short":"S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503.","ista":"Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. 45(2), 467–503.","ama":"Henheik SJ. Deformational rigidity of integrable metrics on the torus. <i>Ergodic Theory and Dynamical Systems</i>. 2025;45(2):467-503. doi:<a href=\"https://doi.org/10.1017/etds.2024.48\">10.1017/etds.2024.48</a>","chicago":"Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the Torus.” <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/etds.2024.48\">https://doi.org/10.1017/etds.2024.48</a>.","mla":"Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the Torus.” <i>Ergodic Theory and Dynamical Systems</i>, vol. 45, no. 2, Cambridge University Press, 2025, pp. 467–503, doi:<a href=\"https://doi.org/10.1017/etds.2024.48\">10.1017/etds.2024.48</a>."},"doi":"10.1017/etds.2024.48","month":"02","issue":"2","ddc":["510"],"quality_controlled":"1","date_updated":"2026-04-07T12:37:10Z","_id":"18112","type":"journal_article","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_type":"hybrid","date_published":"2025-02-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","publisher":"Cambridge University Press","intvolume":"        45","ec_funded":1,"isi":1,"related_material":{"record":[{"id":"19540","status":"public","relation":"dissertation_contains"}]},"project":[{"grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d"},{"_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","name":"Spectral rigidity and integrability for billiards and geodesic flows","grant_number":"885707","call_identifier":"H2020"}],"publication_identifier":{"eissn":["1469-4417"],"issn":["0143-3857"]},"volume":45,"day":"01","language":[{"iso":"eng"}],"publication_status":"published","corr_author":"1","publication":"Ergodic Theory and Dynamical Systems","external_id":{"isi":["001308182000001"]}},{"publisher":"Cambridge University Press","intvolume":"        32","page":"159-165","keyword":["Applied Mathematics","General Mathematics"],"publication_identifier":{"issn":["0143-3857","1469-4417"]},"abstract":[{"text":"In this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.","lang":"eng"}],"volume":32,"date_created":"2020-09-18T10:47:33Z","day":"01","extern":"1","language":[{"iso":"eng"}],"article_type":"original","title":"A Cr unimodal map with an arbitrary fast growth of the number of periodic points","publication_status":"published","publication":"Ergodic Theory and Dynamical Systems","author":[{"first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","orcid":"0000-0002-6051-2628","last_name":"Kaloshin","full_name":"Kaloshin, Vadim"},{"full_name":"KOZLOVSKI, O. S.","last_name":"KOZLOVSKI","first_name":"O. S."}],"citation":{"mla":"Kaloshin, Vadim, and O. S. KOZLOVSKI. “A Cr Unimodal Map with an Arbitrary Fast Growth of the Number of Periodic Points.” <i>Ergodic Theory and Dynamical Systems</i>, vol. 32, no. 1, Cambridge University Press, 2012, pp. 159–65, doi:<a href=\"https://doi.org/10.1017/s0143385710000817\">10.1017/s0143385710000817</a>.","ama":"Kaloshin V, KOZLOVSKI OS. A Cr unimodal map with an arbitrary fast growth of the number of periodic points. <i>Ergodic Theory and Dynamical Systems</i>. 2012;32(1):159-165. doi:<a href=\"https://doi.org/10.1017/s0143385710000817\">10.1017/s0143385710000817</a>","chicago":"Kaloshin, Vadim, and O. S. KOZLOVSKI. “A Cr Unimodal Map with an Arbitrary Fast Growth of the Number of Periodic Points.” <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press, 2012. <a href=\"https://doi.org/10.1017/s0143385710000817\">https://doi.org/10.1017/s0143385710000817</a>.","ista":"Kaloshin V, KOZLOVSKI OS. 2012. A Cr unimodal map with an arbitrary fast growth of the number of periodic points. Ergodic Theory and Dynamical Systems. 32(1), 159–165.","short":"V. Kaloshin, O.S. KOZLOVSKI, Ergodic Theory and Dynamical Systems 32 (2012) 159–165.","ieee":"V. Kaloshin and O. S. KOZLOVSKI, “A Cr unimodal map with an arbitrary fast growth of the number of periodic points,” <i>Ergodic Theory and Dynamical Systems</i>, vol. 32, no. 1. Cambridge University Press, pp. 159–165, 2012.","apa":"Kaloshin, V., &#38; KOZLOVSKI, O. S. (2012). A Cr unimodal map with an arbitrary fast growth of the number of periodic points. <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0143385710000817\">https://doi.org/10.1017/s0143385710000817</a>"},"doi":"10.1017/s0143385710000817","month":"02","issue":"1","_id":"8504","quality_controlled":"1","date_updated":"2021-01-12T08:19:44Z","oa_version":"None","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","type":"journal_article","date_published":"2012-02-01T00:00:00Z","year":"2012","article_processing_charge":"No"},{"status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","year":"2006","date_published":"2006-06-01T00:00:00Z","article_processing_charge":"No","month":"06","doi":"10.1017/s0143385705000714","issue":"3","_id":"8514","date_updated":"2021-01-12T08:19:48Z","quality_controlled":"1","oa_version":"None","day":"01","extern":"1","article_type":"original","language":[{"iso":"eng"}],"title":"The effect of projections on fractal sets and measures in Banach spaces","publication_status":"published","citation":{"ieee":"W. OTT, B. HUNT, and V. Kaloshin, “The effect of projections on fractal sets and measures in Banach spaces,” <i>Ergodic Theory and Dynamical Systems</i>, vol. 26, no. 3. Cambridge University Press, pp. 869–891, 2006.","apa":"OTT, W., HUNT, B., &#38; Kaloshin, V. (2006). The effect of projections on fractal sets and measures in Banach spaces. <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0143385705000714\">https://doi.org/10.1017/s0143385705000714</a>","mla":"OTT, WILLIAM, et al. “The Effect of Projections on Fractal Sets and Measures in Banach Spaces.” <i>Ergodic Theory and Dynamical Systems</i>, vol. 26, no. 3, Cambridge University Press, 2006, pp. 869–91, doi:<a href=\"https://doi.org/10.1017/s0143385705000714\">10.1017/s0143385705000714</a>.","ista":"OTT W, HUNT B, Kaloshin V. 2006. The effect of projections on fractal sets and measures in Banach spaces. Ergodic Theory and Dynamical Systems. 26(3), 869–891.","short":"W. OTT, B. HUNT, V. Kaloshin, Ergodic Theory and Dynamical Systems 26 (2006) 869–891.","ama":"OTT W, HUNT B, Kaloshin V. The effect of projections on fractal sets and measures in Banach spaces. <i>Ergodic Theory and Dynamical Systems</i>. 2006;26(3):869-891. doi:<a href=\"https://doi.org/10.1017/s0143385705000714\">10.1017/s0143385705000714</a>","chicago":"OTT, WILLIAM, BRIAN HUNT, and Vadim Kaloshin. “The Effect of Projections on Fractal Sets and Measures in Banach Spaces.” <i>Ergodic Theory and Dynamical Systems</i>. Cambridge University Press, 2006. <a href=\"https://doi.org/10.1017/s0143385705000714\">https://doi.org/10.1017/s0143385705000714</a>."},"publication":"Ergodic Theory and Dynamical Systems","author":[{"full_name":"OTT, WILLIAM","last_name":"OTT","first_name":"WILLIAM"},{"first_name":"BRIAN","last_name":"HUNT","full_name":"HUNT, BRIAN"},{"full_name":"Kaloshin, Vadim","last_name":"Kaloshin","orcid":"0000-0002-6051-2628","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","first_name":"Vadim"}],"publisher":"Cambridge University Press","page":"869-891","intvolume":"        26","abstract":[{"text":"We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275). More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness exponent $\\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally) Lipschitz functions from $B$ to $\\mathbb{R}^{m}$ that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function $f \\in M$, the Hausdorff dimension of $f(X)$ is at least $\\min\\{ m, d / (1 + \\tau) \\}$. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on $X$. The factor $1 / (1 + \\tau)$ can be improved to $1 / (1 + \\tau / 2)$ if $B$ is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when $\\tau = 0$. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case $\\tau > 0$.","lang":"eng"}],"volume":26,"publication_identifier":{"issn":["0143-3857","1469-4417"]},"date_created":"2020-09-18T10:48:52Z"}]