engleski

Fractal Properties 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 $f$. The phase and amplitude of oscillatory integrals can depend also on additional parameters. The phase could have either nondegenerate or degenerate critical points, depending on the value of the parameters. The caustic is a hypersurface in the parameter space that is the set of all values of the parameters such that the phase has degenerate critical points. Finally, we show an example of a family of caustics that undergoes a bifurcation, which can be seen using the fractal properties approach. Used techniques include Newton diagrams and the resolution of singularities. The Newton diagram technique is commonly used in the analysis of vector fields and maps, and also for the bifurcation analysis.

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

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