Dynamical phase transition for a logarithm observable in chaotic systems

We consider the full distribution $P_N(A)$ of the dynamical observable $A = - \sum_{i=1}^N \ln x_i$ where $x_1, \dots, x_N$ is a sequence generated by a chaotic map $f(x)$ via $x_{n+1} = f(x_n)$. We study the particular cases where $f(x)$ is the doubling, tent or logistic map, and focus on the limit