Singular continuous ergodic measures for the map $z \to z^2$

Question

Where can I find the details of constructing singular continuous ergodic measures for the map $z \to z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it explicitly in his book.

Answer 1

$z \mapsto z^2$ is conjugated to $x \mapsto 2x$ mod 1 on ${\bf R}/{\bf Z}$. This map is in turn semiconjugated to the shift map $\sigma(\{a_n\}_{n\in {\bf N}} = \{a_{n+1}\}_{n\in {\bf N}}$ on $\{0,1\}^{\bf N}$ through base 2 decomposition. $$ \{a_n\} \mapsto \sum_n {a_n \over 2^{n+1}} $$ So you can push any ergodic probability measure invariant by the shift to get a continuous measure which is singular from Lebesgue as soon as it is not Lebesgue. So for example, consider for some $p \in (0,1)$ not equal to $1/2$, $$ \mu_p = (p\delta_{0} + (1-p)\delta_{1})^{\otimes {\bf N}} $$ and push this measure to the circle with the map $$ \matrix{\varphi : & \{0,1\}^{\bf N} & \longrightarrow & S^1 \cr &\{a_n\} & \mapsto & exp\Bigl(2\pi i \sum_n {a_n \over 2^{n+1}}\Bigr) \cr} $$ There are many ergodic probability measures invariant by the shift which have full support, far more than just the $\mu_p$. They are built using thermodynamic formalism. A reference is the book of Bowen, LNM 470, Equilibrium states for Axiom A diffeomorphisms. Any book about hyperbolic or symbolic dynamics should fit. Brin Stuck, Introduction to dynamical systems, Katok-Hasselblatt Modern theory of dynamical systems etc.