Context. The asteroseismology of rapidly rotating pulsating stars is hindered by our poor knowledge of the effect of the rotation on the oscillation properties. Aims. Here we present an asymptotic analysis of high-frequency acoustic modes in rapidly rotating stars. Methods. We study the Hamiltonian dynamics of acoustic rays in uniformly rotating polytropic stars and show that the phase space structure has a mixed character, with regions of chaotic trajectories coexisting with stable structures like island chains or invariant tori. To interpret the ray dynamics in terms of acoustic mode properties, we then use tools and concepts developed in the context of quantum physics. Results. Accordingly, the high-frequency acoustic spectrum is a superposition of frequency subsets associated with dynamically independent phase space regions. The subspectra associated with stable structures are regular and can be modelled through EBK quantization methods, while those associated with chaotic regions are irregular but with generic statistical properties. The results of this asymptotic analysis are successfully compared with the properties of numerically computed high-frequency acoustic modes. The implications for the asteroseismology of rapidly rotating stars are discussed. Key words: hydrodynamics - waves - chaos - stars: oscillations - stars: rotation
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