Finite pattern problems in infinite iterated function systems

Let { f n } n ≥ 1 be an infinite iterated function system on [ 0 , 1 ] and let Λ be its attractor. Then, any x ∈ Λ corresponds to a sequence of integers { a n ( x ) } n ≥ 1 , called the digit sequence of x , in the sense that x = lim n → ∞ ⁡ f a 1 ( x ) ∘ ⋯ ∘ f a n ( x ) ( 1 ) . In this note, let G be a countable set of functions from N to N satisfying g ( n ) → + ∞ as n → + ∞ for every g ∈ G . We investigate the size of the set consisting of numbers whose digit sequences in the infinite iterated function system are strictly increasing and contain every finite pattern from G . We prove that for any infinite iterated function system satisfying the d -decaying condition, the Hausdorff dimension of such a set is 1 / d .