The multifractal spectra of certain planar recurrence sets in the continued fraction dynamical system

In this paper, we will introduce and study the planar recurrence level setS(α,β):={(x,y)∈[0,1)2:R̲(x,y)=α,R¯(x,y)=β},0≤α≤β≤∞, where R̲(x,y) and R¯(x,y) are the lower and upper quantitative hitting time indicators of y by x respectively in the continued fraction dynamical system. It is shown that the multifractal spectrum is always degenerate in a sense that it is constant independent of the choice of the level set.