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Archive of Issues

Uzbekistan Tashkent
Year
2020
Volume
30
Issue
3
Pages
343-366
 Section Mathematics Title The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break Author(-s) Dzhalilov A.A.a, Karimov J.J.ab Affiliations Turin Polytechnic University in Tashkenta, National University of Uzbekistanb Abstract Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, be a circle homeomorphism with one break point $x_{b}$, at which $T'(x)$ has a discontinuity of the first kind and both one-sided derivatives at the point $x_{b}$ are strictly positive. Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction beginning from a certain place coincides with the golden mean, i.e., $\rho_{T}=[m_{1}, m_{2}, \ldots, m_{l}, \, m_{l + 1}, \ldots]$, $m_{s} = 1$, $s> l> 0$. Since the rotation number is irrational, the map $T$ is strictly ergodic, that is, possesses a unique probability invariant measure $\mu_{T}$. A.A. Dzhalilov and K.M. Khanin proved that the probability invariant measure $\mu_{G}$ of any circle homeomorphism $G \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break point $x_{b}$ and the irrational rotation number $\rho_{G}$ is singular with respect to the Lebesgue measure $\lambda$ on the circle, i.e., there is a measurable subset of $A \subset S^{1}$ such that $\mu_ {G} (A) = 1$ and $\lambda (A) = 0$. We will construct a thermodynamic formalism for homeomorphisms $T_{b} \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break at the point $x_{b}$ and rotation number equal to the golden mean, i.e., $\rho_{T}:= \frac {\sqrt{5} -1}{2}$. Using the constructed thermodynamic formalism, we study the exponents of singularity of the invariant measure $\mu_{T}$ of homeomorphism $T$. Keywords circle homeomorphism, break point, rotation number, invariant measure, thermodynamic formalism UDC 517.9 MSC 37A05, 28D05 DOI 10.35634/vm200301 Received 24 February 2020 Language Russian Citation Dzhalilov A.A., Karimov J.J. The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 3, pp. 343-366. References Arnol'd V.I. Small denominators. I. Mappings of the circumference onto itself, Am. Math. Soc., Transl., II. Ser., 1965, vol. 46, pp. 213-284. https://zbmath.org/?q=an:0152.41905 Bowen R. Metody simvolicheskoi dinamiki (Methods of symbolic dynamics), Moscow: Mir, 1979. Dzhalilov A.A., Khanin K.M. On an invariant measure for homeomorphisms of a circle with a point of break, Functional Analysis and Its Applications, 1998, vol. 32, no. 3, pp. 153-161. https://doi.org/10.1007/BF02463336 Dzhalilov A.A. The Hölder property of singular invariant measures of circle homeomorphisms with single corners, Theoretical and Mathematical Physics, 1999, vol. 121, no. 3, pp. 1557-1566. https://doi.org/10.1007/BF02557202 Dzhalilov A.A. Thermodynamic formalism and singular invariant measures for critical circle maps, Theoretical and Mathematical Physics, 2003, vol. 134, no. 2, pp. 166-180. https://doi.org/10.1023/A:1022271903129 Dzhalilov A.A. Limiting laws for entrance times of critical mappings of a circle, Theoretical and Mathematical Physics, 2004, vol. 138, no. 2, pp. 190-207. https://doi.org/10.1023/B:TAMP.0000014851.67668.fb Cornfeld I.P., Fomin S.V., Sinai Ya.G. Ergodic theory, New York: Springer, 1982. https://doi.org/10.1007/978-1-4615-6927-5 Sinai Ya.G., Khanin K.M. Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russian Mathematical Surveys, 1989, vol. 44, no. 1, pp. 69-99. https://doi.org/10.1070/RM1989v044n01ABEH002008 Cunha K., Smania D. Rigidity for piecewise smooth homeomorphisms on the circle, Advances in Mathematics, 2014, vol. 250, pp. 193-226. https://doi.org/10.1016/j.aim.2013.09.017 Denjoy A. Sur les courbes définies par les équations différentielles à la surface du tore, Journal de Mathématiques Pures et Appliquées, 1932, vol. 11, pp. 333-376. de Faria E., de Melo W. Rigidity of critical circle mappings I, Journal of the European Mathematical Society, 1999, vol. 1, issue 4, pp. 339-392. https://doi.org/10.1007/s100970050011 Herman M.R. Sur la conjugaison différentiable des difféomorphismes du cercle a des rotations, Publications Mathématiques de l'Institut des Hautes études Scientifiques, 1979, vol. 49, issue 1, pp. 5-233. https://doi.org/10.1007/BF02684798 Katznelson Y., Ornstein D. The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory and Dynamical Systems, 1989, vol. 9, issue 4, pp. 643-680. https://doi.org/10.1017/S0143385700005277 Khanin K.M., Khmelev D. Renormalizations and rigidity theory for circle homeomorphisms with singularities of break type, Communications in Mathematical Physics, 2003, vol. 235, no. 1, pp. 69-124. https://doi.org/10.1007/s00220-003-0809-5 Khanin K., Kocić S. Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks, Geometric and Functional Analysis, 2014, vol. 24, issue 6, pp. 2002-2028. https://doi.org/10.1007/s00039-014-0309-0 Marmi S., Moussa P., Yoccoz J.-C. Linearization of generalized interval exchange maps, Annals of Mathematics, 2012, vol. 176, no. 3, pp. 1583-1646. http://doi.org/10.4007/annals.2012.176.3.5 de Melo W., van Strien S. One-dimensional dynamics, Berlin: Springer, 1993. https://doi.org/10.1007/978-3-642-78043-1 Ruelle D. Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics, Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511617546 Sinai Ya.G. Gibbs measures in ergodic theory, Russian Mathematical Surveys, 1972, vol. 27, no. 4, pp. 21-69. https://doi.org/10.1070/RM1972v027n04ABEH001383 Vul E.B., Sinai Ya.G., Khanin K.M. Feigenbaum universality and the thermodynamic formalism, Russian Mathematical Surveys, 1984, vol. 39, no. 3, pp. 1-40. https://doi.org/10.1070/RM1984v039n03ABEH003162 Vul E.B., Khanin K.M. Circle homeomorphisms with weak discontinuities, Advances in Sov. Math, 1991, vol. 3, pp. 57-98. https://bookstore.ams.org/advsov-3 Yoccoz J.-C. Conjugaison différentiable des difféomorphismes du cercle dont le nomber de rotation vérifie une condition diophantienne, Annales scientifiques de l'école Normale Supérieure, Serie 4, 1984, vol. 17, no. 3, pp. 333-359. https://doi.org/10.24033/asens.1475 Full text