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.

 

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: