## We have the following distortion estimate

Since the derivative varies too fast in the right neighborhood of the singularity, inspired by [2], we partition the interval (0,δ)(0,δ) into subintervals on which Df(x)Df(x) IU1 inhibitor comparable as follows, where δ is sufficiently small and without loss of generality we let δ=e−Δδ=e−Δ, Δ∈NΔ∈N: for μ≥Δμ≥Δ, μ∈Nμ∈N, let Iμ=(e−(μ+1),e−μ)Iμ=(e−(μ+1),e−μ) and further partition each IμIμ into μ2μ2 subintervals called Iμj Iμj of equal length. For convenience, we also refer to [f2(0),0)[f2(0),0) and (δ,f(0)](δ,f(0)] as IμjIμj intervals. Denote this partition as QQ.

For x,y∈Ix,y∈I, let [x,y][x,y] be the interval connecting x and y. We say x and y have the same itinerary (with respect to the above partition QQ) before time n if fk([x,y])fk([x,y]) does not contain more than three IμjIμj's for 0≤k<n0≤k<n.

Notations: For simplification, in the proof of the following Proposition 6, Corollary 7 and Proposition 9, we use C to denote a positive constant, whose value may vary from expression to expression.