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Finite data rigidity for one-dimensional expanding maps
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 45 / Issue 5 / May 2025
- Published online by Cambridge University Press:
- 13 November 2024, pp. 1597-1618
- Print publication:
- May 2025
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- Article
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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.
