Let G be an end of a Riemann surface with null boundary and let G ′ be a lacunary end with a closed set F = G - G ′ . We study minimal functions in G and G ′ to show that G and G ′ have similar properties if F is thinly distributed on the ideal boundary. We discuss the behaviour of analytic functions in G ′ and relation between the existence of analytic functions of some classes in G ′ and the structure of Martin’s boundary points over the end G . Also we show that the existence of complicated Martin’s boundary points allows only violent analytic functions to exist in G ′ , if F is very thin at the ideal boundary of R .