Investigation of the dynamics of some horseshoe maps which are modificated

Dynamic systems are one of the important research topics that have applications in many disciplines. In particular, investigating whether a dynamical system is chaotic gives us important informations about the dynamical system. The most common definition of chaos in the literature is given by Devaney. In this definition, sensitive dependence on initial conditions, density of periodic points and topological transitivity are examined. In this thesis, Shift map, Folding map, Tent maps on the interval [0, 1] and Cantor set and Smale's Horseshoe map, which are the most basic examples of dynamical systems, are examined in detail and their periodic points are investigated by obtaining the expressions of these structures in binary and triple system. Basically, after the Smale's Horseshoe map is discussed in detail, a new mapping, which we call the first modification of Smale's Horseshoe, is defined, inspired by this map, and then this defined transformation is expressed in ternary system, and then its periodic points are calculated. It is shown in detail that the modified Horseshoe map is a chaotic dynamical system in the Devaney sense on C ×C. In addition, we obtain a map that we call the second modification of Smale's Horseshoe, which is topologically conjugate to the first modifications of Smale's Horseshoe map. Finally, a different modification of Smale's Horseshoe map is obtained and some periodic points are calculated. And then we show that Smale's Horseshoe map, first and second modifications of Smale's Horseshoe map are not topological conjugate to the last modification.