Sierpinski Gasket: Find Three IFS Maps and an Orbit Distance Bound

Begin with an equilateral triangle $T_0$ of side length $1$, with vertices $A$, $B$, and $C$. Remove the open central triangle whose vertices are the side midpoints. Repeat this removal in each surviving corner triangle. The remaining limit set is the Sierpinski gasket $G$.

(a) Specify three maps $f_1,f_2,f_3$ that produce this construction as an iterated function system.

(b) Starting at $x_0$ inside $T_0$, let $x_k$ be the point reached after $k$ iterations. Give an upper bound for its distance to $G$, and state the exponential convergence rate.