engleski

Fractal Geometry of Oscillatory Integrals and Singularities of Differentiable Maps

It is well known that theory of singularities is closely related to the study of asymptotic of oscillatory integrals. We investigate the fractal properties of a geometrical representation of oscillatory integrals \[ I(\tau)=\int_{; ; \mathbb{; ; R}; ; ^n}; ; e^{; ; i\tau f(x)}; ; \phi(x) dx, \] for large values of a real parameter $\tau$, where $f$ is the analytic phase and $\phi$ is the smooth amplitude with compact support. We are motivated by a geometrical representation of Fresnel oscillatory integrals by a spiral called the clothoid, and the idea to produce a classification of singularities using the fractal dimension. We measure the oscillatority by the Minkowski dimension of the planar curve parametrized by the real part $X$ and imaginary part $Y$ of the integral $I$. Also, we measure the oscillatory dimension that is defined as the Minkowski dimension of the graph of the function $x(t) = X(1/t)$, near $t=0$, and analogously for $Y$. We provide explicit formulas connecting these Minkowski dimensions and associated Minkowski contents with asymptotics of the integral $I$ and the type of the critical point of the phase $f$. Used techniques include Newton diagrams and the resolution of singularities. Finally, it is possible to consider oscillatory integrals whose amplitude $\phi$ is a discontinuous function, which is related to more involved fractal geometry.

oscillatory integral; box dimension; Minkowski content; critical points; Newton diagram

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