On the regularity of de Rham curves
On the regularity of de Rham curves
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论德拉姆曲线的规律性
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
V. Protasov
中科院分区：
文献类型：
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作者：
V. Protasov
德？拉姆曲线从一个？多边形弧通过在反复切断角的限制：在每一步，弧段被分成三个部分的比例，其中a？给定参数。我们发现明确的尖锐指数的规律性，这样一个？曲线对于任何？正则性是理解在自然参数化使用弧长作为？参数.我们还获得了一个？a的局部正则性的公式de？Rham曲线，并描述具有给定局部正则性的点集.特别是，我们的特点与最大和最小的局部正则性的点集。平均正则性，这是几乎无处不在的勒贝格措施，计算某些线性算子的李雅普诺夫指数。我们得到了平均正则性的积分公式，并得到了上界和下界。
De?Rham curves are obtained from a?polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio , where is a?given parameter. We find explicitly the sharp exponent of regularity of such a?curve for any?. Regularity is understood in the natural parametrization using the arclength as a?parameter. We also obtain a?formula for the local regularity of a?de?Rham curve at each point and describe the sets of points with given local regularity. In particular, we characterize the sets of points with the largest and the smallest local regularity. The average regularity, which is attained almost everywhere in the Lebesgue measure, is computed in terms of the Lyapunov exponent of certain linear operators. We obtain an integral formula for the average regularity and derive upper and lower bounds.