Let $\varphi$ be a homeomorphism on a totally disconnected, compact metric space $X$. We introduce a binary relation on the family of clopen subsets of $X$, which is described in terms of the $\varphi$-invariant probability measures. We show that $\varphi$ is uniquely ergodic if and only if any two clopen subsets of $X$ are comparable with respect to the binary relation.

The Cauchy problem for a wave equation with a time periodic coefficient is considered. We prove that if one of the initial data is a compactly supported smooth function and the other initial data is zero, then the energy of the solution of the Cauchy problem grows exponentially. This result is proved by applying the unstable properties of Hill’s equation.

The separability of certain seminormed spaces of harmonic functions on Riemann surfaces will be considered. An application of the result obtained in the above to some inverse inclusion problem in the classification theory of Riemann surfaces will be appended.

In this paper we study energy growth for solutions to wave equations. We prove that there exist compact in space perturbations of the wave equation $\partial_{t}^{2}u-\triangle u=0$ such that the energy of solution grows at the rate $\exp((1+t)^{\alpha})$ for any $\alpha \geq 0$.

Let $U$ be an integer with $U>1$. If $n$ is even with $n\geq 6$, then the class number of $\mathbf{Q}(\sqrt{1-4U^{n}})$ is divisible by $n$ except $(U,n)=(13,8)$.

CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero with three regular embedded ends) in hyperbolic 3-space $H^{3}$ are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so here we give an explicit description of CMC-1 trinoids in $H^{3}$ that includes the reducible case.