We study the classical problem of noisy constrained
capacity, namely, the capacity of a noisy channel whose input is 
a sequence from a constrained set. As stated in [3] ``... while 
calculation of the noise-free capacity of constrained sequences is
well known, the computation of the capacity of a constraint in
the presence of noise . . . has been an unsolved problem in the
half-century since Shannon’s landmark paper . . ..'' We express   
the constrained capacity of a binary symmetric channel with     
$(d, k)$-constrained input as a limit of the top Lyapunov exponents
of certain matrix random processes. We compute asymptotic
approximations of the noisy constrained capacity for cases where
the noise is small. In particular, we show that when $k \le 2d$,
the error term with respect to the constraint capacity is $O(\epsilon)$,
whereas it is $O(\epsilon \log \epsilon)$ when $k > 2d$. 
In both cases, we compute
the coefcient of the error term. We also extend previous results
on the entropy of a hidden Markov process to higher-order finite
memory processes.