Geodesic
Limiting Laws for Entrance Times of Critical Mappings of a Circle
Teoretičeskaâ i matematičeskaâ fizika, Tome 138 (2004) no. 2, pp. 225-245 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A renormalization group transformation $\mathbf R_1$ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number $\rho={(\sqrt{5}-1)}/{2}$ (“the golden mean”). Let a homeomorphism $T$ be the $C^{1}$-conjugate of $T_{\xi_{0},\eta_{0}}$. We let $\{\Phi_n^{(k)}(t), \ n=\overline{1,\infty}\}$ denote the sequence of distribution functions of the time of the $k$th entrance to the $n$th renormalization interval for the homeomorphism $T$. We prove that for any $t\in\mathbb{R}^1$, the sequence $\{\Phi_n^{(1)}(t)\}$ has a finite limiting distribution function $\Phi_n^{(1)}(t)$, which is continuous in $\mathbb{R}^1$, and singular on the interval $[0,1]$. We also study the sequence $\bigl\{\Phi_{n}^{(k)}(t), \ n=\overline{1,\infty}\bigr\}$ for $k>1$.
Keywords: critical homeomorphism of a circle, distribution function of the entrance time, thermodynamic formalism. 
@article{TMF_2004_138_2_a3,
     author = {A. A. Dzhalilov},
     title = {Limiting {Laws} for {Entrance} {Times} of {Critical} {Mappings} of {a~Circle}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {225--245},
     year = {2004},
     volume = {138},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2004_138_2_a3/}
}
TY  - JOUR
AU  - A. A. Dzhalilov
TI  - Limiting Laws for Entrance Times of Critical Mappings of a Circle
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2004
SP  - 225
EP  - 245
VL  - 138
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2004_138_2_a3/
LA  - ru
ID  - TMF_2004_138_2_a3
ER  - 
%0 Journal Article
%A A. A. Dzhalilov
%T Limiting Laws for Entrance Times of Critical Mappings of a Circle
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2004
%P 225-245
%V 138
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2004_138_2_a3/
%G ru
%F TMF_2004_138_2_a3
A. A. Dzhalilov. Limiting Laws for Entrance Times of Critical Mappings of a Circle. Teoretičeskaâ i matematičeskaâ fizika, Tome 138 (2004) no. 2, pp. 225-245. http://geodesic.mathdoc.fr/item/TMF_2004_138_2_a3/

[1] B. Pitskel, Ergodic Theory Dynam. Syst., 11 (1991), 501–513 | DOI | MR | Zbl

[2] M. Hirata, Ergodic Theory Dynam. Syst., 13 (1993), 533–556 | DOI | MR | Zbl

[3] A. Chabra, R. Jensen, Phys. Rev. Lett., 62 (1989), 1327–1340 | DOI | MR

[4] Z. Coelho, de E. Faria, Israel J. Math., 93 (1996), 93–112 | DOI | MR | Zbl

[5] Ya. G. Sinai, Sovremennye problemy ergodicheskoi teorii, Izd. firma “Fiziko-matematicheskaya literatura”, M., 1995

[6] J. C. Yoccoz, Ann. Sci. Ecole Norm. Sup. (4), 17 (1984), 333–359 | DOI | MR | Zbl

[7] K. M. Khanin, Chaos, 1 (1991), 181–186 | DOI | MR | Zbl

[8] R. Ostlund, D. Rand, J. Sethna, E. Siggia, Phys. D, 8 (1983), 303–342 | DOI | MR | Zbl

[9] J.-P. Eckmann, H. Epstein, Commun. Math. Phys., 107 (1986), 213–231 | DOI | MR | Zbl

[10] E. de Faria, W. de Melo, J. Eur. Math. Soc., 1 (1999), 339–392 | DOI | MR | Zbl

[11] A. A. Dzhalilov, TMF, 134:2 (2003), 191–206 | DOI | MR | Zbl

[12] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Classical Equlibrium. Statistical Mechanics, Encyclopedia of Math. and its Applications, 5, Addison-Wesley, Reading, MA, 1978 | MR | Zbl

[13] R. Vouen, Metody simvolicheskoi dinamiki, Mir, M., 1979

[14] A. A. Dzhalilov, TMF, 121:3 (1999), 355–366 | DOI | MR | Zbl