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.