Assouad type dimensions of generalized affine fractal interpolation functions and their applications

Abstract

In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function $W$, whose graph $Γ_W$ has the box dimension $2+\log_Nλ$, we obtain \[ \dim_A^θ(Γ_W) \leq \frac{2+\log_Nλ-θ}{1-θ}, \qquad θ\in \left(0,\log_N\frac{1}λ\right). \] and if $λ^2 N <1$, we get \[ \dim_A(Γ_W)\geq 1 +\log_N\left(\frac{1}λ\right). \] For the classical Takagi function $T$ with graph $Γ_T$, we show that \[ \dim_A^θ(Γ_T)=1, θ\in(0,1), \] and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.