## Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

#### Article Title

### The entrance times for circle maps with a break

#### Abstract

In present work we study the entrance times for circle homeomorphisms with one break point and universal renormalization properties. Consider the set * X * of all orientation preserving circle homeomorphisms *T* with one break point and golden mean rotation number. It is well known that the renormalization group transformation has a unique periodic point * T _{b}* with period 2. Denote by

*B*the set of all circle maps

*C*-conjugated to

^{1 }*T*. Consider the map

_{ b}*T ∈ B*and its unique probability invariant measure

*μ*. Denote by

*E(x)*the first entrance times of

*x*to interval defined by generalized dynamical partition. Consider the rescaled first entrance time. We study convergence in law of random variables of rescaled first entrance time.

#### First Page

209

#### Last Page

221

#### References

1. Sinai Ya.G. Topics in Ergodic Theory. Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ. 1994.

2. Chunha K., Smania D. Renormalization for piecewise smooth homeomorphisms on the circle. Preprint: arXiv 1108.1968 (2011).

3. Chunha K., Smania D. Rigidity for piecewise smooth homeomorphisms on the circle. Preprint: arXiv 1201.1401 (2012).

4. Coelho Z., de Faria E. Limit laws of entrance times for homeomorphisms. Israel J. Math., 93, 93–112 (1996).

5. Coelho Z. The Loss of Tightness of Time Distributions for Homeomorphisms of the Circle. Transactions of the American Mathematical Society, Vol. 356, No.11, 4427–4445 (2004).

6. Collet P., Galves A. Asymptotic distribution of entrance times for expanding maps of an interval. Preprint, (1992).

7. Dzhalilov A.A., Khanin K.M. On an Invariant Measure for Homeomorphisms of the Circle with a Single Break Point. Russian Math. Surveys, Vol. 51, No.6, 1198–1199 (1996).

8. 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, Vol. 32, No.3, 153–161 (1998).

9. Dzhalilov A.A. Limiting Laws for Entrance Times of Critical Mappings of a Circle. Theoret. and Math. Phys., 138:2, 190-207 (2004).

10. Dzhalilov A.A., Liousse I. Circle homeomorphisms with two break points. Nonlinearity, Vol. 19. No.8, 1951–1968 (2006).

11. Dzhalilov A.A., Liousse I. and Mayer D. Singular Measures of Piecewise Smooth Circle Homeomorphisms with two break points. Discrete and Continuous Dynamical Systems, Series-A, No. 2(24), 381–403 (2009).

12. Dzhalilov A., Mayer D. and Safarov U. Piecewise-smooth circle homeomorphisms with several break points. Izvestiya RAN. Ser. Mat., Vol. 76, No.1, 95–113 (2012).

13. Dzhalilov A., Mayer D., Djalilov S. and Aliyev A. An Extention of Herman's Theorem for Nonlinear Circle Maps with Two Breaks. Russian Journal of Nonlinear Dynamics, Vol. 14, No.4, 553-–577, (2018).

14. Hirata M. Poisson law for Axiom A diffeomorphisms. Ergodic Theory Dynamical Systems, Vol. 13, 533–556 (1993).

15. Karimov J.J. On continuity of limit distribution function for rescaled hitting times. Uzbek Mathematical Journal, No.4, 78–88 (2019).

16. Khanin K. M., Vul E.B. Circle Homeomorphisms with Weak Discontinuities. Advances in Sov. Math. 3, 57–-98 (1991).

17. Khanin K., Kocic S. Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks. Geom. Funct. Anal. 24, 2002–-2028 (2014).

18. Kim D.H., Seo B.K. The waiting time for or rational rotations. Nonlinearity 16, 1861–1868 (2003).

19. Pitskel B. Poisson limit law for Markov chains. Ergodic Theory Dynamical Systems, Vol. 11, 501–513 (1991).

20. Teplinskii A.Yu., Khanin K. M. Rigidity for circle diffeomorphisms with singularities. Uspekhi Mat. Nauk, 59, No.2(356), 137–160 (2004).

#### Recommended Citation

Dzhalilov, Akhtam and Karimov, Javlon
(2020)
"The entrance times for circle maps with a break,"
*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*: Vol. 3:
Iss.
2, Article 10.

DOI: https://doi.org/10.56017/2181-1318.1097