Dimensions of a class of nonautonomous carpets and measures on $\R^{2}$

Yifei Gu; Chuanyan Hou; Jun Jie Miao

doi:10.4171/jfg/158

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Dimensions of a class of nonautonomous carpets and measures on $R^{2}$

Yifei Gu
East China Normal University, Shanghai, P. R. China
Chuanyan Hou
Xinjiang Normal University, Urumqi, Xinjiang, P. R. China
Jun Jie Miao
East China Normal University, Shanghai, P. R. China

$Dimensions of a class of nonautonomous carpets and measures on $\R^{2}$ cover$

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Abstract

For each integer $k > 0$ , let $n_{k}$ and $m_{k}$ be integers such that $n_{k} \geq 2,$ $m_{k} \geq 2$ , and let $D_{k}$ be a subset of ${0, \dots, n_{k} - 1} \times {0, \dots, m_{k} - 1}$ . For each $w = (i, j) \in D_{k}$ , we define an affine transformation on $R^{2}$ by

Φ_{w} (x) = T_{k} (x + w), w \in D_{k},

where $T_{k} = diag (n_{k}^{- 1}, m_{k}^{- 1})$ . The non-empty compact set

E = k = 1 ⋂ \infty (w_{1} w_{2} \dots w_{k}) \in \prod_{i = 1}^{k} D_{i} ⋃ Φ_{w_{1}} \circ Φ_{w_{2}} \circ \dots \circ Φ_{w_{k}}

is called a nonautonomous carpet.
In the paper, we provide the lower, packing, box-counting and Assouad dimensions of the nonautonomous carpets $E$ . We also explore the dimension properties of nonautonomous measures $μ$ supported on $E$ , and we provide Hausdorff, packing and entropy dimension formulas of $μ$ .

Cite this article

Yifei Gu, Chuanyan Hou, Jun Jie Miao, Dimensions of a class of nonautonomous carpets and measures on $R^{2}$ . J. Fractal Geom. (2024), published online first

DOI 10.4171/JFG/158