The loss of tightness of time distributions for homeomorphisms of the circle
Author:
Zaqueu Coelho
Translated by:
Journal:
Trans. Amer. Math. Soc. 356 (2004), 44274445
MSC (2000):
Primary 37E05, 11A55, 37A50; Secondary 28D05, 60G55
Published electronically:
February 4, 2004
MathSciNet review:
2067127
Fulltext PDF Free Access
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Abstract: For a minimal circle homeomorphism we study convergence in law of rescaled hitting time point process of an interval of length . Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence going to zero exhibiting coexistence of two nontrivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.
 [Bre]
Leo
Breiman, Probability, AddisonWesley Publishing Company,
Reading, Mass.LondonDon Mills, Ont., 1968. MR 0229267
(37 #4841)
 [BZ]
Xavier
Bressaud and Roland
Zweimüller, Non exponential law of entrance times in
asymptotically rare events for intermittent maps with infinite invariant
measure, Ann. Henri Poincaré 2 (2001),
no. 3, 501–512 (English, with English and French summaries). MR 1846853
(2002f:37018), http://dx.doi.org/10.1007/PL00001042
 [Coe]
Zaqueu
Coelho, Asymptotic laws for symbolic dynamical systems, Topics
in symbolic dynamics and applications (Temuco, 1997) London Math. Soc.
Lecture Note Ser., vol. 279, Cambridge Univ. Press, Cambridge, 2000,
pp. 123–165. MR 1776758
(2001h:37016)
 [CC]
Zaqueu
Coelho and Pierre
Collet, Asymptotic limit law for the close approach of two
trajectories in expanding maps of the circle, Probab. Theory Related
Fields 99 (1994), no. 2, 237–250. MR 1278884
(95g:60034), http://dx.doi.org/10.1007/BF01199024
 [CF]
Zaqueu
Coelho and Edson
de Faria, Limit laws of entrance times for homeomorphisms of the
circle, Israel J. Math. 93 (1996), 93–112. MR 1380635
(97e:58141), http://dx.doi.org/10.1007/BF02761095
 [CG]
P.
Collet and A.
Galves, Asymptotic distribution of entrance times for expanding
maps of the interval, Dynamical systems and applications, World Sci.
Ser. Appl. Anal., vol. 4, World Sci. Publ., River Edge, NJ, 1995,
pp. 139–152. MR 1372959
(97b:58083), http://dx.doi.org/10.1142/9789812796417_0011
 [DM]
Fabien
Durand and Alejandro
Maass, Limit laws of entrance times for lowcomplexity Cantor
minimal systems, Nonlinearity 14 (2001), no. 4,
683–700. MR 1837633
(2002f:37001), http://dx.doi.org/10.1088/09517715/14/4/302
 [DV]
D.
J. Daley and D.
VereJones, An introduction to the theory of point processes,
Springer Series in Statistics, SpringerVerlag, New York, 1988. MR 950166
(90e:60060)
 [FHZ1]
S.
Ferenczi, C.
Holton, and L.
Q. Zamboni, Structure of three interval exchange transformations.
I. An arithmetic study, Ann. Inst. Fourier (Grenoble)
51 (2001), no. 4, 861–901 (English, with
English and French summaries). MR 1849209
(2002g:11116)
 [FHZ2]
S. Ferenczi, C. Holton & L. Zamboni, Combinatorics of threeinterval exchanges, Proceedings ICALP 2001, Lecture Notes in Computer Science 2076 (2001), SpringerVerlag, 567578.
 [FHZ3]
S. Ferenczi, C. Holton & L. Zamboni, Structure of threeinterval exchange transformations II: combinatorial description of the trajectories, J. Analyse Math 89 (2003), 239276.
 [FHZ4]
S. Ferenczi, C. Holton & L. Zamboni, Structure of threeinterval exchange transformations III: ergodic and spectral properties, submitted.
 [FHZ5]
S. Ferenczi, C. Holton & L. Zamboni, Joinings of threeinterval exchange transformations, submitted.
 [GS]
G.
R. Grimmett and D.
R. Stirzaker, Probability and random processes, 2nd ed., The
Clarendon Press, Oxford University Press, New York, 1992. MR 1199812
(93m:60002)
 [Hir]
Masaki
Hirata, Poisson law for Axiom A diffeomorphisms, Ergodic
Theory Dynam. Systems 13 (1993), no. 3,
533–556. MR 1245828
(94m:58137), http://dx.doi.org/10.1017/S0143385700007513
 [HSV]
Masaki
Hirata, Benoît
Saussol, and Sandro
Vaienti, Statistics of return times: a general framework and new
applications, Comm. Math. Phys. 206 (1999),
no. 1, 33–55. MR 1736991
(2001c:37007), http://dx.doi.org/10.1007/s002200050697
 [Nev]
J.
Neveu, Processus ponctuels, École
d’Été de Probabilités de SaintFlour,
VI—1976, SpringerVerlag, Berlin, 1977, pp. 249–445.
Lecture Notes in Math., Vol. 598 (French). MR 0474493
(57 #14132)
 [Pit]
B.
Pitskel′, Poisson limit law for Markov chains, Ergodic
Theory Dynam. Systems 11 (1991), no. 3,
501–513. MR 1125886
(92j:60081), http://dx.doi.org/10.1017/S0143385700006301
 [Bre]
 L. Breiman, Probability theory, AddisonWesley Series in Statistics, 1968. MR 37:4841
 [BZ]
 X. Bressaud & R. Zweimüller, Non exponential law of entrance times in asymptotically rare events for intermittent maps with infinite invariant measure, Annales Henri Poincaré 2, No.3 (2001), 501512. MR 2002f:37018
 [Coe]
 Z. Coelho, Asymptotic laws for symbolic dynamical systems, in Topics in Symbolic Dynamics and Applications, Eds. F. Blanchard, A. Maass & A. Nogueira, LMS Lecture Notes Series 279, Cambridge Univ. Press, (2000) 123165. MR 2001h:37016
 [CC]
 Z. Coelho & P. Collet, Limit law for the close approach of two trajectories of expanding maps of the circle, Prob. Th. and rel. fields 99 (1994), 237250. MR 95g:60034
 [CF]
 Z. Coelho & E. de Faria, Limit laws of entrance times for homeomorphisms of the circle, Israel J. Math. 93 (1996), 93112. MR 97e:58141
 [CG]
 P. Collet & A. Galves, Asymptotic distribution of entrance times for expanding maps of the interval, Dynamical Systems and Applications 139152, World Sci. Ser. Appl. Anal., 4, World Sci. Publishing, 1995. MR 97b:58083
 [DM]
 F. Durand & A. Maass, Limit laws of entrance times for low complexity Cantor minimal systems, Nonlinearity 14, No.4 (2001), 683700. MR 2002f:37001
 [DV]
 D.J. Daley & D. VereJones, An introduction to the theor y of point processes, Springer Series in Statistics, SpringerVerlag, 1988. MR 90e:60060
 [FHZ1]
 S. Ferenczi, C. Holton & L. Zamboni, Structure of threeinterval exchange transformations I: an arithmetic study, Ann. Inst. Fourier 51, 4 (2001), 861901. MR 2002g:11116
 [FHZ2]
 S. Ferenczi, C. Holton & L. Zamboni, Combinatorics of threeinterval exchanges, Proceedings ICALP 2001, Lecture Notes in Computer Science 2076 (2001), SpringerVerlag, 567578.
 [FHZ3]
 S. Ferenczi, C. Holton & L. Zamboni, Structure of threeinterval exchange transformations II: combinatorial description of the trajectories, J. Analyse Math 89 (2003), 239276.
 [FHZ4]
 S. Ferenczi, C. Holton & L. Zamboni, Structure of threeinterval exchange transformations III: ergodic and spectral properties, submitted.
 [FHZ5]
 S. Ferenczi, C. Holton & L. Zamboni, Joinings of threeinterval exchange transformations, submitted.
 [GS]
 G.R. Grimmett & D.R. Stirzaker, Probability and Random Processes, Clarendon Press, Oxford, 1992. MR 93m:60002
 [Hir]
 M. Hirata, Poisson law for Axiom A diffeomorphisms, Erg. Th. Dyn. Sys 13 (1993), 533556. MR 94m:58137
 [HSV]
 M. Hirata, B. Saussol & S. Vaienti, Statistics of return times: a general framework and new applications, Comm. Math. Physics 206 (1999), 3355. MR 2001c:37007
 [Nev]
 J. Neveu, Processus pontuels, in Springer Lecture Notes in Maths 598 (1976), 249445. MR 57:14132
 [Pit]
 B. Pitskel, Poisson limit law for Markov chains, Erg. Th. Dyn. Sys. 11 (1991), 501513. MR 92j:60081
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Additional Information
Zaqueu Coelho
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom
Email:
zc3@york.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002994704033860
PII:
S 00029947(04)033860
Keywords:
Rotation numbers,
limit laws,
point processes,
maps of the circle
Received by editor(s):
August 31, 2001
Received by editor(s) in revised form:
May 8, 2003
Published electronically:
February 4, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
