Summary: Let \(f\) be a transcendental meromorphic function and denote by \(J(f)\) the Julia set and by \(I(f)\) the escaping set. We show that if \(f\) has a direct singularity over infinity, then \(I(f)\) has an unbounded component and \(I(f)\cap J(f)\) contains continua. Moreover, under this hypothesis \(I(f)\cap J(f)\) has an unbounded component if and only if \(f\) has no Baker wandering domain. If \(f\) has a logarithmic singularity over infinity, then the upper box dimension of \(I(f)\cap J(f)\) is 2 and the Hausdorff dimension of \(J(f)\) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have ’direct or logarithmic tracts’, but which need not be meromorphic in the plane. These results are obtained by using a generalization of the Wiman-Valiron theory. This method is also applied to complex differential equations.