Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and $\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a more standard statistical terminology would be covariance function) is defined by
$$C_n(\varphi ,\phi,T) = \int_X (\varphi \circ T^n)\phi~ d\mu - \int_X \varphi ~d\mu \int_X \phi ~d\mu.$$
The transformation $T$ is mixing if for any $\varphi,\phi \in L^2$, $C_n(\varphi ,\phi,T) \to 0$ as $n\to \infty$.
A natural question in Thermodynamic Formalism is to ask the speed of convergence of a mixing transformation. Without extra assumptions, the convergence can be arbitrarily slow. So the question is a little different, what is the correlation decay speed for a collection of functions $\mathcal{L}\subsetneq L^2$. An interesting case is when we get an exponential decay, that occur in spaces such as Hölder continuous, Zygmund or bounded variation functions. pPolynomial decay can indicate some intermittence in the system.
So, given a mixing transformation, we look for a suitable Banach space $\mathcal{L}$ to analyze the decay rate of the correlations. Here is my question about these studies, normally we look for a Hölder-like space, I believe this has its roots in physics, but why is this a good space to analyze the decay rate? It looks like we're forcing to find a space with exponential decay, for example. Why, does a given space of observables provide useful information about system, are there any results talking about it?
Does the presence of exponential decay give us any topological information about the system? I ask this because, as I said, the decay rate will depend on the reference space. I've already found some works that relate this stochastic property of mixing with topological properties, but in this case, it doesn't depend on the decay rate, just that the transformation is mixing.
If anyone has any reference on this question, I would appreciate it, but it seems to me that the choice of Hölder-like spaces is related to the roots of thermodynamic formalism in physics, I could be wrong, but that's why I'm asking here.