The real decay rate

Let f be a C^2 uniform expanding map on the 1-torus \mathbb{T}, \mu be the unique ACIP of f, which is exponentially mixing. That is, there exists \lambda\in(0,1) such that |C(\phi,\psi,f^n)|\le C\lambda^n for any two Lipschitz functions \phi,\psi on \mathbb{T}, where
C(\phi,\psi,f^n,\mu)=\int \phi\circ f^n\cdot \psi d\mu -\mu(\phi)\cdot\mu(\psi) be the correlation function.

Let h be a C^2 diffeomorphism on \mathbb{T}, g=h^{-1}fh be the induced map, and h_\ast \nu=\mu. The new correlation function

C(\phi,\psi,g^n,\nu)=\int \phi\circ g^n\cdot \psi d\nu-\nu(\phi)\cdot\nu(\psi)
=\int \hat\phi\circ f^n(hx)\cdot \hat\psi(hx) d\nu-\nu(\phi)\cdot\nu(\psi)
=\int \hat\phi\circ f^n \cdot \hat\psi d\mu-\mu(\hat\phi)\cdot\mu(\hat\psi),

where \hat\phi=\phi\circ h^{-1}. Therefore, the two smoothly conjugate systems (g,\nu) and (f,\mu) have the same mixing rate.

Assuming h is close to identity, we see that g=h^{-1}fh is also C^2 uniformly expanding, and one may derive the mixing rate of (g,\nu) independently. However, this new rate may be different (better or worse) from \lambda. For example, f could be the linear expanding ones and archive the best possible rate among its conjugate class. Could one detect this rate from (g,\nu) itself?

In the general case, two expanding maps on \mathbb{T} are only topologically conjugate (via full shifts). So it is possible that the decay rate varies in the topologically conjugate classes.


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