Functions and Logarithms

Algebra Level 4

$\begin{aligned} f(x) & = & 10^{10x} \\ g(x) & = & \log_{10} \left ( \frac x {10} \right ) \\ h_1 (x) & = & g(f(x)) \\ h_n (x) & = & h_1 (h_{n-1} (x)) \\ \end{aligned}$

Denote the set functions above. Evaluate the sum of digits of $h_{2011} (1)$ .

Note- This is not original. Adapted from a Math contest.