clarification of definition of ifs

Question

Could any-one tell me what they meant by dots here for both the places used? especially with probabilities?

Answer 1

The first dots in the random orbit mean $$T_{j_n} \circ T_{j_{n-1}} \circ T_{j_{n-2}} \circ \dots \circ T_{j_1}(x)$$ since we iteratively apply the maps of the IFS. The probabilities are $$p_{j_1}(x) \cdot \prod_{k=2}^n p_{j_k}(T_{j_{k-1}} \circ \dots \circ T_{j_1}(x))$$ and these stand for the probabilities in the iterated application: We first apply $T_{j_1}$ to $x$ with probability $p_{j_1}(x)$. After that we arrive at $T_{j_1}(x)$ and we apply $T_{j_2}$ to $T_{j_1}(x)$ with probability $p_{j_2}(T_{j_1}(x))$, so altogether we apply $T_{j_2}$ after $T_{j_1}$ to $x$ with probability $$p_{j_1}(x) \cdot p_{j_2}(T_{j_1}(x))$$

by the formula for conditional probability. An inductive approach now leads to the claimed probabilities.