Hausdorff dimension of the Rauzy gasket
Hausdorff dimension of the Rauzy gasket
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发表时间：
2023-12
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通讯作者：
Natalia Jurga
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作者：
Natalia Jurga
Rauzy 垫片是射影平面上抛物线非共形迭代函数系统的吸引子，它描述了各种重要拓扑和动力学问题中的特殊参数集。自 2009 年以来，人们多次尝试计算 Rauzy 垫片的 Hausdorff 维数，由于几何形状中抛物线性和非共形性的结合，这是一个具有挑战性的问题。在本文中，我们通过证明 Rauzy 垫片的豪斯多夫维数等于（投影）亲和维数来解决这个问题。支持这一点的关键技术成果是将 Hochman 和 Solomyak 的工作部分推广到 $\mathrm{SL}_3(\mathbb{R})$ 设置，其中我们确定了 Rauzy 垫圈上支持的固定（Furstenberg）测量的精确尺寸。固定测度和吸引子的维度结果都建立在更广泛的普遍性上，并将最近关于射影迭代函数系统的工作扩展到更高的维度。
The Rauzy gasket is the attractor of a parabolic, nonconformal iterated function system on the projective plane which describes an exceptional parameter set in various important topological and dynamical problems. Since 2009 there have been several attempts to calculate the Hausdorff dimension of the Rauzy gasket, which is a challenging problem due to the combination of the parabolicity and nonconformality in the geometry. In this paper we settle this question by proving that the Hausdorff dimension of the Rauzy gasket equals the (projective) affinity dimension. The key technical result underpinning this is a partial generalisation of work of Hochman and Solomyak to the $\mathrm{SL}_3(\mathbb{R})$ setting, where we establish the exact dimension of stationary (Furstenberg) measures supported on the Rauzy gasket. The dimension results for both stationary measures and attractors are established in broader generality and extend recent work on projective iterated function systems to higher dimensions.