General properties of the Ruelle operator

Question

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics.

I have been interested in some technical properties of the Ruelle-Perron-Frobenius operator, RPF operator for short.

The RPF operator appears fist in the context of the Statistical Mechanics, in a seminal paper of Ruelle. After that it was use in Dynamical systems in a very successful way.

Nowadays, I'm investigating properties of the RPF operator in some general contexts. Then I'm interested in understand some technical details of the proof of some properties of the RPF operator.

To remember, the RPF operator acts over the space of the continuous functions $C^{0}(X, \mathbb{C})$, where $X=\{0,1\}^{\mathbb{N}}$ is endowed with the product topology (or equivalently with the topology generated by the metric $d_\theta(x,y)=\theta^i, i=\min\{k,x_k\neq y_k\}$). For a fixed $f\in C^{0}(X,\mathbb{C})$ we define the RPF operator $L_f:C^{0}(X,\mathbb{C})\to C^{0}(X,\mathbb{C})$ by putting $$ L_f\varphi(x)=\sum_{y\in \sigma^{-1}x}\varphi(y)e^{f(y)}. $$

In our context we shall suppose that $f=u+iv$, $u$ is a normalized potential, i.e. $L_u1=1$, $m$ is a fixed point for the dual operator $L_u^{*}$. $F_\theta$ denote the subspace of the Lipschitz functions, with the metric $d_\theta$.

Fix a potential $f=u+iv$. Consider the operator $\mathcal{V}$ defined by $\mathcal{V}(\varphi)=e^{iv}\cdot (\varphi \circ \sigma).$ The operator $\mathcal{V}$ acts over $L^{2}(m), C^{0}(X)$ and $F_\theta$.

My problem: I'm trying to understand Proposition 4.2 of chapter 4 of (1) which is:

Prop. $ \mathcal{V}$ has an $L^2(m)$ eigenfunction iff $\mathcal{V}$ has an $F_{\theta}$ eigenfunction.

I can understand almost of the proof given in (1) but I can't understand a little detail:

Claim: If $(\varphi\circ \sigma^n)/\varphi=\alpha^n (e^{iv})^n$ then $\varphi L_u^n(\dfrac{g}{\varphi})=\alpha^nL_f^ng$

I can proof this claim for $n=1$ and I show the left hand for all $n$, in fact is sufficiently multiply $(\varphi\circ \sigma^n)/\varphi$ by $g$ and apply $L_u$ in order to get $\varphi L_u^n(\dfrac{g}{\varphi})$ if $n$ equals 1 then the same trick works to the right hand. But for $n\geq 2$ I can't apply the same trick for the right side.

Another claim that I can't proof is in proposition 4.3, says that $\int \varphi L_f^ng dm=\int (\mathcal{V}^n\varphi)g dm.$ I think that in both of them we must use in some way that $L_u1=1$ but I can't realize.

Answer 1

I rather prefer the notation $S_n w,$ instead of $w_n.$

Proof of claim 1: $ \varphi\circ \sigma^n/\varphi=\alpha^n e^{iS_nv},$ then $$ \varphi(x) L_u^n(g/\varphi)(x)=\sum_{y\in\sigma^{-n}x}(\varphi\circ\sigma^n (y)/\varphi(y)) g(y) e^{S_n u(y)}=\sum_{y\in\sigma^{-n}x}\alpha^n e^{iS_nv} g(y) e^{S_n u(y)}\\=\alpha^n\sum_{y\in\sigma^{-n}x} g(y) e^{S_n f(y)}, $$ i.e. $\varphi L_u^n(g/\varphi)=\alpha^n L_f^n g,$ as claimed.

I can now explain you what happens...The proof of prof 4.2 starts by ...then $w\circ\sigma e^{-iv}=\alpha w.$ It follows saying that ... $w\circ \sigma^n / w=\alpha^ne^{iv^n},$ however, if you write down this, you will realise that indeed $w\circ \sigma^n / w=\alpha^ne^{iS_nv}.$ Therefore, you will realise that in their notation $v^n$ should be replaced by $v_n.$

With respect to your second claim, in the proof of Prop 4.3, it is not used that $L_u 1=1,$ neither that $V(\varphi)=e^{iv}(\varphi\circ \sigma).$ The only fact that are being used are the basic relationship between $L_f$ and $V$ (5th line, second page of CH4), some properties of the Banach space $\mathcal{L}^2(m),$ and the conclusion of Prop 4.1 (not its proof).