Research and Communications in Mathematics and Mathematical Sciences Vol. 5, Issue 1, 2015, Pages 11-32 ISSN 2319-6939 Published Online on May 08, 2015  2015 Jyoti Academic Press http://jyotiacademicpress.net 2010 Mathematics Subject Classification: 37E10, 37B20, 37N25. Keywords and phrases: circle homeomorphism, displacement sequence, integrate-and-fire models. Communicated by Lixin Cheng. Received January 29, 2015 ON THE REGULARITY OF THE DISPLACEMENT SEQUENCE OF AN ORIENTATION PRESERVING CIRCLE HOMEOMORPHISM WACŁAW MARZANTOWICZ 1 and JUSTYNA SIGNERSKA 2 1 Faculty of Mathematics and Computer Science Adam Mickiewicz University of Poznań ul. Umultowska 87 61-614 Poznań Poland e-mail: marzan@amu.edu.pl 2 Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8 00-956 Warszawa Poland e-mail: j.signerska@impan.pl Abstract We investigate the regularity properties of the displacement sequence () () () ( ) , 2 exp , 1 mod 1 ix z x x z n n n π = Φ − Φ = η − where R R → Φ : is a lift of an orientation preserving circle homeomorphism. If the rotation number () q p = ϕ  is rational, then () z n η is asymptotically periodic with semi-period q. This