论文标题
在贝德福德·麦克马伦地毯的中间尺寸上
On the intermediate dimensions of Bedford-McMullen carpets
论文作者
论文摘要
$ \dim_θλ$表示的其他地方的中间维度，使用参数$θ\在[0,1] $中的hausdorff和盒子尺寸之间进行了插值。确定$ \dim_θλ$的精确公式特别具有挑战性，当$λ$是带有独特的Hausdorff和盒子尺寸的Bedford-McMullen地毯。在这个方向上，回答弗雷泽的问题，我们表明$ \dim_θλ$严格小于每个$θ<1 $的$λ$的盒子尺寸，此外，上限的派生范围严格呈$θ= 1 $。我们还改进了Falconer，Fraser和Kempton获得的下限。
The intermediate dimensions of a set $Λ$, elsewhere denoted by $\dim_θΛ$, interpolates between its Hausdorff and box dimensions using the parameter $θ\in[0,1]$. Determining a precise formula for $\dim_θΛ$ is particularly challenging when $Λ$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this direction, answering a question of Fraser, we show that $\dim_θΛ$ is strictly less than the box dimension of $Λ$ for every $θ<1$, moreover, the derivative of the upper bound is strictly positive at $θ=1$. We also improve on the lower bound obtained by Falconer, Fraser and Kempton.
