Entropy and frame measures for self-similar measures

Abstract

For $0<ρ<1$ and a set of $N\ge2$ distinct real digits $\mathscr D=\{d_0,\ldots,d_{N-1}\}\subset\mathbb{R}$, let \[ μ_{ρ,\mathscr D}=\frac1N\sum_{i=0}^{N-1}μ_{ρ,\mathscr D}\circ S_i^{-1}, \qquad S_i(x)=ρ(x+d_i). \] Write $X=\sum_{j\ge1}ρ^j d_{ω_j}$, where the symbols $ω_j$ are independent and uniform on $\{0,\ldots,N-1\}$. We prove that $μ_{ρ,\mathscr D}$ admits no frame measure if the conditional entropy $H(ω_1\mid X)>0$. In particular, this holds for all $ρ>1/N$, without any absolute-continuity assumption or arithmetic restriction on the digits or the contraction. At $ρ=1/N$, for commensurable digits, a frame measure exists if and only if $μ_{ρ,\mathscr D}$ is normalized Lebesgue measure on its attractor. For $\mathscr D=\{0,\ldots,N-1\}$ and $ρ<1/N$, we also prove nonexistence whenever $ρ^{-m}=a/b$ for some $m\in\N$, with $a/b$ in lowest terms, $b\mid N$ and $N/b\nmid a$, extending the integer-power obstruction. The extension compares mask quotients using finite families of target pole classes in place of one target class.