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I was trying to solve an exercise about the Manneville-Pomeau example, but I got stuck. I need to give a little background before posting the exercise. Thank you guys in advance for your attention. I'm sorry, I know the question is quite huge but I'm really lost here.

Let $\alpha>0$ be a real number and let $a$ be the only number in $(0,1)$ such that $a(1+a^{\alpha})=1$. Define $f:[0,1]\rightarrow[0,1]$ in the following way

$f(x)=x(1+x^{\alpha})$ if $x\in[0,a]$ and $f(x)=\frac{x-a}{1-a}$ if $x\in(a,1]$.

Let $\{a_n\}_n\subset [0,a]=:E$ be defined as $a_0=1$, $a_1=a$ and $a_n=f(a_{n+1})$. Consider $g(x)=f^{\rho(x)}(x)$ where $$\rho(x)=1+\min\{n\geq0: f^n(x)\in(a,1]\}$$

Define $$\nu_\rho(B)= \sum_{n=0}^\infty \sum_{k>n} m(f^{-n}(B)\cap E_k)$$ where $B\subset[0,1]$, $m$ is the Lebesgue measure and $E_k$ is such that $\rho(x)=k$ for all $x\in E_k$.

I already know that $\nu_\rho$ is invariant for $f$ and also that $\nu_\rho(M)=\int_{E}\rho dm$. In particular, $\nu_\rho$ is finite iff $\rho\in L_1(m)$.

Here are the problems I can't solve: Prove that $\nu_\rho$ is $\sigma$-finite. Show that $(a_n)$ is monotone decreasing and converges to 0. Also, there exists $c_1, c_2>0$ such that

$c_m\leq a_jj^{1/\alpha}\leq c_2$ and $c_m\leq (a_j-a_{j+1})j^{1+(1/\alpha)}\leq c_2$ for all j

Conclude that the $g$-invariant measure $\nu_\rho$ is finite iff $\alpha \in (0,1)$.

Thank you guys so much for the attention and for the time! Thanks in advance for all future help and I'm sorry again for the huge post!!

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