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Tropical dynamics of area-preserving maps

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  • We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.

    Mathematics Subject Classification: Primary: 37P50; Secondary: 37E30.

    Citation:

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  • Figure 1.  Orbits of a hyperbolic map on three randomly constructed tropical K3 surfaces

    Figure 2.  Pictures of currents in Rubik's cube example

    Figure 3.  Stable and unstable currents of a perturbed Kummer example, viewed from different angles. The perturbed Kummers exhibit tangency of the stable and unstable manifolds

    Figure 4.  Stable and unstable currents in the Kummer examples have smooth potentials and are uniformly hyperbolic

    Figure 5.  Left: The monomials that are minimized in each region of the plane, together with the tropical elliptic curve in the $ (e_1, e_2) $-plane. Right: The dual subdivision of the Newton polytope. The Legendre transform of the function on the left determines the subdivision on the right. In the picture, all affine linear functions in the definition of $ h $ are minimized for some value of $ e_1, e_2 $

    Figure 6.  A tropical elliptic curve with the skeleton in bold and dashed reflection lines. The dotted vertical and horizontal lines denote the points where the reflection lines change slope

    Figure 7.  The iterate of a segment under the twist map, an analogue of §6.2 in the present case

    Figure 8.  Left: The invariant curves of the rotation and the lines of reflection for the involutions. Right: The break lines of the function h° which determines the core pencil

    Figure 9.  A fundamental domain in the $(a, b)$ plane $\mathbb{R}^2$ for the $\mathbb{Z}^2$ and $\pm 1$ action, and its image under the map to $\mathbb{R}^3$. The domain is divided into $4$ triangles where the embedding is affine, with corresponding affine maps to $\mathbb{R}^3$ indicated on each triangle. The face and equations of the image tetrahedron are:
    $ABC: x+y-z+1 = 0 \quad \quad BCD: -(x+y+z) + 1 = 0$
    $ABD: x-y+z+1 = 0 \quad \quad ACD: -x+y+z + 1 = 0$

    Figure 10.  Typical pictures at the corners of a tropical K3 surface

    Figure 11.  Tropical K3 surfaces in the Rubik's cube family. Left: level set in $[\frac{1}{2}, 1]$. Right: level set $>1$. The surfaces are not drawn to scale, i.e. in the $\mathbb{R}^3$ that contains both, the one on the left is much smaller

    Figure 12.  Forward (red) and backward (blue) iterates of the triangle face on the tropical K3. Left: for a small value of t. Right: for a large value of t. Figure 2 contains further examples of iterates of the triangle face for a Rubik's cube example for large t

    Figure 13.  The lifted tent map, and its action on the section σ

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