Consider the map $f(x)=3^x$ mod 1. Using the the iterated function system $T_{0}x=\log_{3}(x+1), T_{1}x=\log_{3}(x+2)$ we see that $f$ is dynamical conjugated to a full shift on two symbols. Moreover i think it follows from Lasota and York (Trans AMS, 1973) that there is an absolutely continuous ergodic measure for $f$. What is the density of this measure? Is this measure Bernoulli or Markov (an image of a Bernoulli or Markov measure under the conjugation)? What is the entropy of this measure?