On Invariant Measures of the Space of Sequences Associated by Circle Maps with Irrational Rotation Number
Keywords:
circle homeomorphism, break point, rotation number, invariant measure, symbolic dynamics, left shift map, thermodynamic formalism, transfer matrix operator, renormalization transformationAbstract
Let $\mathcal{A}=\{a,0,1\}$ be an alphabet. Consider the space
$$\Theta_{+}:=\{\underline{a}=(a_{1},a_{2},\dots a_{n},\dots),\,\,\,{{a}_{n}}\in \mathcal{A},\,\,\, n=\overline{1,\infty},\,\,\,$$
$$a_{n+1}=a,\,\,\, \textrm{if and only if, }\,\,\ a_{n}=0\}.$$
Define the shift $\sigma:\Omega\rightarrow \Omega$ by $\sigma(\underline{a})_i=a_{i+1}$, $i\geq 1$. Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_{b}\}),\,\,\,\varepsilon>0,$ be a circle homeomorphism with a single break point $ x_{b}$ and irrational rotation number $\rho_{T}=\frac{-k+\sqrt{{{k}^{2}}+4}}{2},\,\,k\ge 1$. We build probability invariant measure $\mu$ with respect to shift $\sigma$.
References
Hadamard J. Les surfaces a courbures opposees et leurs lignes geodesiques. J. Math. Pures Appl. 4, pages 27–73, 1898.
Arnold V.I. Small denominators. i. mapping the circle onto itself. Izvestiya Akademii Nauk SSSR, issue 25(1), pages 21–86, 1961.
Herman M.R. Sur la conjugaison differentiable des dif-feomorphismes du cercle a des rotations. Publications Mathematiques de l’IHES, Volume 49, pages 5–233, 1979.
Yoccoz J.C. Conjugaison differentiable des diffeomor-phismes du cercle dont le nomber de rotation verifie une condition diophantienne. Ann.Sci.Ecole Norm. Sup., No 17(3), pages 333–359, 1984.
Khanin K.M. Sinai Y. G. Smoothness of conjugacies of diffeomorphisms of the circle with rotations. Russian Mathematical Surveys, No. 44(1), page 69, 1989.
Ornstein D. Katznelson Y. The differentiability of the conjugation of certain diffeomorphisms of the circle. Ergodic Theory Dynamical Systems, No. 9(4), pages 643–680, 1989.
Khanin K.M. Dzhalilov A.A. On an invariant measure for homeomorphisms of a circle with a point of break. Funct. Anal. Its Appl., No. 32, pages 153–161, 1998.
Liousse I. Pl homeomorphisms of the circle which are piecewise c1 conjugate to irrational rotations. Bull Braz. Math. Soc., New Series 35, page 269–280, 2004.
Liousse I. Dzhalilov A.A. Circle homeomorphisms with two break points. Nonlinearity, V. 19. No. 8, pages 1951–1968, 2006.
Mayer D. Dzhalilov A.A., Liousse I. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete and Continuous Dynamical Sys-tems, Series-A, No. 2(24), pages 381–403, 2009.
Safarov U. Dzhalilov A., Mayer D. Piecewise-smooth circle homeomorphisms with several break points. Izvestiya RAN. Ser. Mat., V. 76, No. 1, pages 95–113, 2012.
Sinai Ya.G. Topics in Ergodic Theory. 1994.
Denjoy A. Sur les courbes definies par les equations differentielles a la surface du tore. Math. Pures et Appl., No. 11, pages 333–375, 1932.
Karimov J. J. Dzhalilov A. A. The thermodynamic for-malism and exponents of singularity of invariant mea-sure of circle maps with a single break. Bulletin of Ud-murt University. Mathematics, Mechanics, Computer Science, pages 343–366, 2020.
Karimov J. J. Dzhalilov A. A. On exponents of probability invariant measure of circle maps with a break point. AIP Conference Proceedings 2781, pages 020059–1–020059–6, 2023.
