TITLE: How can generic closed geodesics on surfaces of negative curvature be constructed?
QUESTION [6 upvotes]: As far as I understand it the closing lemma implies that closed geodesics on surfaces of negative curvature are dense. So: how can they be constructed in general? 
A concrete answer that dovetails with the construction of such surfaces with constant negative curvature and genus $g$ from regular hyperbolic $(8g-4)$-gons along lines indicated by Adler and Flatto and gives the endpoints of the geodesics in the Poincaré disk model would be ideal. More useful still would be a way to construct all the closed geodesics that cross the boundaries of translates of the fundamental $(8g-4)$-gon some specified number of times (I am pretty sure this ought to be a finite set, but I couldn't say why off the top of my head).

REPLY [6 votes]: Are you willing to buy that the set of  non-closed geodesics are dense?  If so, here is an argument that goes back to Birkhoff and Hadamard.  Take the surface's 
universal cover -- which is to say the upper half plane.  Tile it with fundamental domains for said surface's fundamental group (relative to a fixed const neg curv metric).  These
have some number of edges, a, b, c, ... . Now count how a geodesic crosses the edges:
acbaf... thus getting an (infinite) word -- or symbol sequence--  in the edges.  Theorem: the symbol sequence  is periodic if and only if the geodesic is. Theorem:  if we are given a (variable) negatively curved metric on the surface
then the symbol sequence uniquely determines the geodesic.  Theorem: if a sequence $s_N$ of 
symbol sequences converges to a symbol sequence $s$, in the sense that for any sufficiently large `window' L of word length centered around 0, the finite word of length L of arbitrary
length eventually agree, then the corresponding geodesics also converge.   Now,
approximate your given geodesic -- ie symbol sequence -- by a periodic sequence. 
symbol sequence by longer and longer periodic sequences.