  In this paper, we study obstructions to the Dirichlet property by two
approaches: density of non-positive points and functionals on adelic
R-divisors. Applied to the algebraic dynamical systems, these results provide
examples of nef adelic arithmetic R-Cartier divisor which does not have the
Dirichlet property. We hope the obstructions obtained in the article will give
ways toward criteria of the Dirichlet property.
