BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS

Let Phi = {phi(i) : i is an element of Lambda} be an iterated function system on a compact metric space (X, d), where the index set Lambda = {1, 2, ..., l} with l >= 2, or Lambda = {1, 2, ...}. We denote by J the attractor of Phi, and by D the subset of points possessing multiple codings. For any x is an element of J\D, there is a unique integer sequence {omega(n)(x)}(n >= 1) subset of Lambda(N), called the digit sequence of x, such that {x} = boolean AND(n) phi(omega 1(x)) omicron ... omicron phi(omega n(x)) (X). In this case we write x = [omega(1)(x), omega(2) (x), ...]. For x, y is an element of J\D, we define the shortest distance function M-n(x, y) as M-n(x, y) = max {k is an element of N : omega(i+1)(x) = omega(i+1)(y), ..., omega(i+k) (x) = omega(i+k) (y) for some 0 <= i <= n - k}, which counts the run length of the longest same block among the first n digits of (x, y). In this paper, we are concerned with the asymptotic behaviour of M-n(x, y) as n tends to infinity. We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance function. As applications, we study the exceptional sets in several concrete systems such as continued fractions system, Luroth system, N-ary system, and triadic Cantor system.

Automated summary

This passage discusses the attractor and points with multiple codings in an iterated function system on a compact metric space. It introduces the concepts of digit sequences and shortest distance functions to describe the relationships between points. The paper focuses on the asymptotic behavior of the shortest distance function as n approaches infinity, calculates the Hausdorff dimensions of exceptional sets, and studies exceptional sets in specific systems.