  We study billiards in domains enclosed by circular polygons. These are closed
$C^1$ strictly convex curves formed by finitely many circular arcs. We prove
the existence of a set in phase space, corresponding to generic sliding
trajectories close enough to the boundary of the domain, in which the return
billiard dynamics is semiconjugate to a transitive subshift on infinitely many
symbols that contains the full $N$-shift as a topological factor for any $N \in
\mathbb{N}$, so it has infinite topological entropy. We prove the existence of
uncountably many asymptotic generic sliding trajectories approaching the
boundary with optimal uniform linear speed, give an explicit exponentially big
(in $q$) lower bound on the number of $q$-periodic trajectories as $q \to
\infty$, and present an unusual property of the length spectrum. Our proofs are
entirely analytical.
