Algorithms generating images of attractors of generalized iterated function systems

Abstract

Abstract. The paper is devoted to searching algorithms which will allow to generate images of attractors of generalized iterated function systems (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space X, we consider mappings form the Cartesian product X × ... × X to X).Two presented algorithms are counterparts of classical deterministic algorithm and so-called chaos game.The third and fourth one one is fitted to special kind of GIFSs -to affine GIFS, which are, in turn, also investigated.

Approximating the Attractor Set of Iterated Function Systems of Order m by $$\alpha $$-Dense Curves

“…The concept of iterated function systems, IFS, to construct fractals was introduced by Hutchinson [12] in the decade 1980-1990, and later popularized by Barnsley [3]. Due to the applications of the fractals in many applied sciences (see, for instance, [7,8,14,29,36] and references therein), the IFS have been widely studied; see [2,6,11,13,15,[17][18][19][20]28,[30][31][32][33][34].…”

Making the computation of approximations of invariant measures and its attractors for IFS and GIFS, through the deterministic algorithm, tractable

“…obtained by the updating rule (3). The first coordinates of D N , given by supp(M N +1 S (µ)) approximate the attractor set A S , and the second coordinates {v 1 , ..., v m }, gives the value at each point of the discrete probability M N S (µ) approximating the invariant probability µ S .…”

“…The deterministic one, where and initial set or measure is iterated by the respective operators approximating the attractor or the invariant measure w.r.t. the appropriated topology, see [1,2] for classical IFS and [3] for GIFS. The discrete one, is similar to the deterministic but an initial step is to introduce a discrete version of the space, an ε-net, and a discrete version of the operator, which is then iterated in the same fashion as the deterministic one producing a discrete set close to the attractor and a discrete measure which is close to the invariant one, [4][5][6].…”

Sensing and Controlling with Markov Process for Locally Independent Fractal Image

“…The IFS is a representative system that can be used as a basis to simulate the scenery of a fractal nature to describe common fractal shapes, and its theory and practice have been widely applied in many fields. (11,12) As the shape of an ordinary IFS fractal attractor is very regular, the attractor often seems to be "rigid" and does not significantly change when the IFS attractor is directly used to simulate a natural landscape. Garcla introduced a mutually recursive function system (MRFS), a popular form of the IFS, to obtain a balance between order and chaos in the IFS attractor image.…”