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Local inversion of a class of piecewise regular maps

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The authors were partially supported by the "Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni" of the Istituto Nazionale di Alta Matematica "F. Severi".
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  • This paper provides sufficient conditions for any map L, that is strongly piecewise linear relatively to a decomposition of $\mathbb{R}^k$ in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the PC1 case.

    Mathematics Subject Classification: 26B10, 47H11, 47J07.

    Citation:

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  • Figure 1.  The image of the unit disk centered at the origin in the plane $z = 0$ under the map $G$ of Example 2.4.

    Figure 2.  The three cones in Example 3.2 for different choices of α and β. The z axis is not shown because it is assumed perpendicular to the page.

    Figure 3.  Image of the unit sphere under $\mathcal{G}$ as in Example 3.2 for different choices of the parameters $\alpha$, $\beta$, $s_1$, $s_2$, $s_3$. The dark continuous line represents the image of the circle of radius $1$ centered at the origin.

    Figure 4.  The image of the unit sphere centered at the origin under the map G of Example 3.5 The green lines on the left are the intersections of the cones C1, …, C4 with the unit sphere, those on the right are their images.

    Figure 5.  The decomposition of $\mathbb{R}^3$ of Example 3.12

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