Finite Data Rigidity for One-Dimensional Expanding Maps

Let $f,g$ be $C^2$ expanding maps on the circle which are topologically conjugate. We assume that the derivatives of $f$ and $g$ at corresponding periodic points coincide for some large period $N$. We show that $f$ and $g$ are "approximately smoothly conjugate." Namely, we construct a $C^2$ conjugacy $h_N$ such that $h_N$ is exponentially close to $h$ in the $C^0$ topology, and $f_N:=h_N^{-1}gh_N$ is exponentially close to $f$ in the $C^1$ topology. Our main tool is a uniform effective version of Bowen's equidistribution of weighted periodic orbits to the equilibrium state.