This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.7 − −1.9 (2D) and 2.4 − −2.7 (3D).1 There are also literature discussing the relationship between turbulence and SLE using random fields [20,21,22], and it is expected that there may be a correspondence between these and our model.

We have shown in [1] that the invariant varieties of periodic points (IVPP) of all periods of some higher dimensional rational maps can be derived, iteratively, from the singularity confinement (SC). We generalize this algorithm, in this paper, to apply to any birational map, which has more invariants than the half of the dimension.

The generalization of squeezing is realized in terms of the Virasoro algebra. The higher-order squeezing can be introduced through the higher-order time-dependent potential, in which the standard squeezing operator is generalized to higher-order Virasoro operators. We give a formula that describes the number of particles generated by the higher-order squeezing when a parameter specifying the degree of squeezing is small. Formula (18) shows that the higher the order of squeezing, the larger the number of generated particles.

We review the classical and quantum theory of the Pais-Uhlenbeck oscillator as the toy-model for quantizing f (R) gravity theories.

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