On the structure of the attractors of hyperbolic iterated function systems (Q1299824)

The author presents the construction of attractors and Markov attractors for hyperbolic iterated function systems. By an iterated function system the author means a compact metric space \(X\) together with a collection of continuous maps \(T_1,\dots,T_n\) on it. If all the \(T_i\)'s are contractions then these iterated function systems are called hyperbolic. The author proves that when the Markov transition matrix \(M\) is an irreducible 0-1 matrix, for a huge amount of elements \((j_1,j_2,\dots)\in\Sigma_{M^t}\) the Markov attractor \(A_M\) can be approached by the set \[ \{T_{j_k}\circ T_{j_{k-1}}\circ\cdots\circ T_{j_1}x,\quad K\geq m\},\quad \forall x\in X. \]