Suppose that $f: X \rightarrow X$ is computable in log-space. Given an input $x \in X$ where $x$ is encoded within $n$ bits, is $f^n(x)$ computable in log-space? 
FL with polynomial number of log-space reductions still in FL?
cc.complexity theory;complexity classes;reductions
The answer is no unless L=P (and obviously yes if L=P).  To see this, fix a deterministic Turing machine (DTM) M which solves a P-complete problem in polynomial time, and consider a function f which maps a configuration of M to the configuration in the next time step.  Function f is just the simulation of one step in the DTM and can be computed in logarithmic space.  If you iterate this function t times, you can compute the configuration after t steps, and therefore you can decide whether DTM M accepts on a given input string within t steps.  If you can do this in logarithmic space, you have solved the P-complete problem in logarithmic space, and therefore L=P.