On properties of similarity boundary of attractors in product dynamical systems

Abstract

<p style='text-indent:20px;'><i>Fractals</i> in higher dimensional <i>dynamical systems</i> have significant roles in physics and other applied sciences. In this paper, one of the key property of fractals, called <i>self similarity</i> in product systems, is studied using the concept of <i>similarity boundary</i>. The relationship between similarity boundary of an attractor in a product space to one of its projection spaces is discussed. The impact of <i>inverse invariance</i> of similarity boundary on its coordinate iterated function system is analyzed. Fractals satisfying the <i>strong open set condition</i>, restricted to attractors in product spaces, are characterized. The relationship between similarity boundary of attractors in product spaces and their overlapping sets is also obtained. The equivalency of the restricted open set condition (ROSC) and the strong open set condition in product spaces, is proved. Self similarity of an attractor in a product system is characterized using the Hausdorff measure of its similarity boundary. Also, the Hausdorff dimensions of the overlapping set and similarity boundary of attractors for different types of iterated function systems are obtained.</p>