普遍曲面的地平线问题
The horizon problem for prevalent surfaces
摘要：
研究了由函数$f \in C[0,1]^2 $定义的分形曲面的视界的盒维数。特别是，我们表明，一个普遍的表面满足“地平线属性”，即盒维数的地平线是一个小于的表面。由于一个普遍曲面的盒维数为3，这并不能给我们任何关于维数严格小于小于3的曲面的视界的信息。为了检验这种情况，我们引入了上盒维数至多为α的曲面的函数空间，对于α $\in$ [2，3]。在这种情况下，视界的行为更加微妙。我们构造了这些空间的一个普遍子集，其中视界的下盒维数介于表面的维数减1和2之间。我们表明，在流行的意义上，这些界限是尽可能紧，如果空间的定义纯粹的尺寸。然而，如果我们在Lipschitz空间中工作，视界性质确实对普遍函数成立。一路上，我们得到了一系列的属性盒维数的函数和。
英文摘要：
We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α$\in$ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
