  We study the density estimation problem with observations generated by
certain dynamical systems that admit a unique underlying invariant Lebesgue
density. Observations drawn from dynamical systems are not independent and
moreover, usual mixing concepts may not be appropriate for measuring the
dependence among these observations. By employing the $\mathcal{C}$-mixing
concept to measure the dependence, we conduct statistical analysis on the
consistency and convergence of the kernel density estimator. Our main results
are as follows: First, we show that with properly chosen bandwidth, the kernel
density estimator is universally consistent under $L_1$-norm; Second, we
establish convergence rates for the estimator with respect to several classes
of dynamical systems under $L_1$-norm. In the analysis, the density function
$f$ is only assumed to be H\"{o}lder continuous which is a weak assumption in
the literature of nonparametric density estimation and also more realistic in
the dynamical system context. Last but not least, we prove that the same
convergence rates of the estimator under $L_\infty$-norm and $L_1$-norm can be
achieved when the density function is H\"{o}lder continuous, compactly
supported and bounded. The bandwidth selection problem of the kernel density
estimator for dynamical system is also discussed in our study via numerical
simulations.
