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Báo cáo hóa học: " FIXED POINTS, PERIODIC POINTS, AND COIN-TOSSING SEQUENCES FOR MAPPINGS DEFINED ON TWO-DIMENSIONAL CELLS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: FIXED POINTS, PERIODIC POINTS, AND COIN-TOSSING SEQUENCES FOR MAPPINGS DEFINED ON TWO-DIMENSIONAL CELLS | FIXED POINTS PERIODIC POINTS AND COIN-TOSSING SEQUENCES FOR MAPPINGS DEFINED ON TWO-DIMENSIONAL CELLS DUCCIO PAPINI AND FABIO ZANOLIN Received 12 January 2004 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 Poincare 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. 1. Introduction and basic settings 1.1. A motivation from the theory of ODEs. This paper deals with the study of fixed points and periodic points as well as with the investigation of chaotic dynamics in a sense that will be described later for continuous maps defined on generalized rectangles of a Hausdorff topological space X. Motivated by the study of the Poincare map associated to some classes of planar ordinary differential systems like equation x y y -w t g x 1.1 which in turn corresponds to the nonlinear scalar Hill equation x w t g x 0 1.2 we introduced in 42 the concept of a map stretching a two-dimensional oriented cell si into another oriented cell . Formally an oriented cell 2ft was defined in 42 as a pair - with Q R2 being the homeomorphic image of a rectangle and with the set - Q dtt playing a role which may remind us but in a very weak sense of that of an exit set in the Conley-WaZewski theory 11 55 56 . The stretching definition was then Copyright 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004 2 2004 113-134 2000 Mathematics Subject Classification 34C25 34C28 37D45 70Kxx URL http dx.doi.org 10.1155