We show that a slow and periodic in time change in the shape of a spatial domain can lead to a non-trivial adiabatic evolution of a quantum-mechanical particle confined in the domain: each period of the perturbation results in a non-trivial permutation of the eigenstates (a bijection N --> N) . The exact permutation rule is determined by the spectrum of the Laplace-Dirichlet operator. The iterations of this permutation are either periodic or escape to infinity. The problem of which of the possibilities is realized for a given permutation seems to be undecidable, but we provide heuristic arguments for the genericity of the escape to infinity with an exponential rate and give an explicit example of such behavior - the exponential energy growth in a periodically connected/disconnected quantum graph.  We also show that one can control the evolution of the state of the quantum-mechanical particle with arbitrarily good precision by variations in the rate of the slow change in the domain shape. This is a joint work with A.Duca and R.Jolie.