Let h be a piecewise-linear (PL) circle homeomorphism with two break points a0 , c0 and irrational rotation number ρh . Denote by qn , n ≥ 1 the first return times of h and 000(0)():(0)hhaahaσ′−=′+ the jump of h at the point a0 . We prove that for every 1xS∈ the sequence 01log()mod1,1log()nqhDhxnaσ≥ is uniformly distributed on [0,1].
 V.I. Arnold, Small denominators I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961) 21–86.  Z. Coelho, A. Lopes and L. da Rocha, Absolutely continuous invariant measures for a class of affine interval exchange maps, Proc. Amer. Math. Soc. 123 (11) (1995) 3533–3542.  I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic Theory, Springer Verlag, Berlin, (1982).  A.A. Dzhalilov and K.M. Khanin, On invariant measure for homeomorphisms of a circle with a point of break, Functional Anal. i Prilozhen. 32 (3) (1998) 11–21, translation in Funct. Anal. Appl. 32 (3) (1998) 153–161.  A.A. Dzhalilov and I. Liousse, Circle homeomorphisms with two break points, Nonlinearity, 19 (2006) p. 1951-1968.  A.A. Dzhalilov, A.A. Jalilov and D. Mayer, A remark on Denjoy’s inequality for P L circle homeomorphisms with two break points, Journal of Mathematical Analysis and Applications, 458 (2018) p. 508-520  A.A. Dzhalilov, D. Mayer and U.A. Safarov, Piecwise-smooth circle homeomorphisms with several break points, Izv. Ross. Akad. Nauk Ser. Mat. 76 (1) (2012) 101–120, translation in Izv. Math. 76 (1) (2012) 94–112.  Y. Katznelson and D. Ornstein, The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor.Dyn.Syst. 9(1989), p.681–690.  M. Herman: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Etudes Sci. Publ. Math. 49 (1979) 5–233. Ergodic Theory Dynamic. Systems 9 (1989) 681–690.  K.M. Khanin and Ya.G. Sinai, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russ. Math. Surveys 44 (1) (1989) 69–99, translation of Uspekhi Mat. Nauk 44 (1) (1989) 57–82.  L.Kuipers and H. Niederreiter: Uniform distribution of sequences. Wiley, New York, 1974.  I. Liousse, P L-Homeomorphisms of the circle which are piecewise C 1 conjugate to irrational rotations,Bull Braz Math Soc, New Series 35(2)(2004), p. 269–280.  H.Nakada, Piecewise linear homeomorphisms of type III and the ergodicity of the cylinder flows, Keio Math. Sem. Rep. N7 (1982), p.29–40.  A. Teplinsky: A circle diffeomorphism with breaks that is smoothly linearizable, Ergod. Ther. and Dynam. Sys. (2018), 38, p.371–383  J.C. Yoccoz, Il n’y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris Sér. I Math. 298 (7) (1984) 141–144.
Dzhalilov, Akhtam and Tashkulov, Khamza
"Uniform Distribution for Piecewise-Linear Herman's Maps with Two Breaks,"
Acta of Turin Polytechnic University in Tashkent: Vol. 8
, Article 2.
Available at: https://uzjournals.edu.uz/actattpu/vol8/iss3/2