Shrinking target problems in d-decaying Gauss-like iterated function systems

Abstract

Let Φ={ϕn}n≥1 be an infinite iterated function system (IFS) on [0,1] satisfying the open set condition with the open set (0,1) and let J be its attractor. For any x∈J, there is a unique integer sequence {ωn(x)}n≥1, called the digit sequence of x, such thatx=limn→∞⁡ϕω1(x)∘⋯∘ϕωn(x)(1) except at countably many points. In this paper, we investigate the shrinking target problem in the infinite iterated function system with a polynomial decay of the derivative. More precisely, we consider the size of the setW(T,x0)={x∈J:Tn(x)∈Itn(x0) for infinitely many n∈N}, where T:J⟼J is the expanding map induced by the left shift and Itn(x0) denotes the tn-th cylinder of x0∈J. As a continuation of one result of Li et al. (2014) [11], we obtain the exact Hausdorff dimension of the set W(T,x0) in some general infinite function systems.