Fixed Point Theory and Applications
Volume 2004 (2004), Issue 2, Pages 113-134

Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells

Duccio Papini1 and Fabio Zanolin2

1Dipartimento dell'Ingegneria dell'Informazione, Università di Siena, Via Roma 56, Siena 53100, Italy
2Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Udine 33100, Italy

Received 12 January 2004

Copyright © 2004 Duccio Papini and Fabio Zanolin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We propose, in the general setting of topological spaces, a definition of two-dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells. For these maps, we prove some theorems on the existence of fixed points, periodic points, and sequences of iterates which are chaotic in a suitable manner. Our results, motivated by the study of the Poincaré map associated to some nonlinear Hill's equations, extend and improve some recent work. The proofs are elementary in the sense that only well known properties of planar sets and maps and a two-dimensional equivalent version of the Brouwer fixed point theorem are used.