2025/04/02 by Rafał Tryniecki, Tryniecki, Rafał, Mariusz Urbański +3
Mathematics · #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2504.02135
openalex publication_date 2025/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G(x):=\1/x\ be the Gauss map. By gn(x)=(1)/(x+n) we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) Gn by limiting the collection of functions gk, k∈\mathbb N, to the first n elements, meaning that Gn = \gk \k=1n. We are interested in the asymptotics of the Hausdorff measure of the limit set Jn i. e. set consisting of irrational elements of [0,1] having continued fraction expansion with entries at most n. In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that lim n → ∞ (1-Hn(Jn))/(1-hn) ⋅ (1)/(ln n) = 1, where Jn is the limit set of the piecewise-linear analogue of Gn, hn is its Hausdorff dimension and Hn is the value of hn-dimensional Hausdorff measure of the set Jn, Hn:=Hhn(Jn). In the second part, we focus on the IFS generated by the first n branches of Gauss map and prove, as our main result, that limn→∞ (1-Hn)/((1-hn)ln n)= 1 and equivalently, due to Hensley's result, limn→∞ (n(1-Hn))/(ln n)= (6)/(π2), where Jn is the limit set of the system Gn, i.e. the set consisting of irrational numbers in [0,1] that continued fraction expansion with entries not exceeding n. Similarly as for the piecewise linear map, hn is the Hausdorff dimension of Jn and Hn is the value of hn-dimensional Hausdorff measure of the set Jn, Hn:=Hhn(Jn).