A three-parametrical family of ODEs on a torus arises from a model of Josephson effect in a resistive case when a Josephson junction is biased by a sinusoidal microwave current. We study asymptotics of Arnold tongues of this family on the parametric plane (the third parameter is fixed) and prove that the boundaries of the tongues are asymptotically close to Bessel functions. ‡ Supported by part by RFBR grants 12-01-31241-mol-a and 12-01-33020-mol-a-ved 1 In physical literature this equation is often written with sines instead of cosines. Substitutions x → x ± π/2, t → t ± π/2 transform one variant to another; one can see that these substitutions do not affect all results of this paper.

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Cesàro convergence of spherical averages is proven for measurepreserving actions of Markov semigroups and groups. Convergence in the mean is established for functions in L p , 1 ≤ p < ∞, and pointwise convergence for functions in L ∞ . In particular, for measure-preserving actions of word hyperbolic groups (in the sense of Gromov) we obtain Cesàro convergence of spherical averages with respect to any symmetric set of generators.