论文标题
脂肪sierpinski垫片上的随机β-转化
Random β-transformation on fat Sierpinski gasket
论文作者
论文摘要
我们考虑迭代功能系统（IFS）$$ f _ {\ vec {q}}（\ vec {z}）= \ frac {\ vec {\ vec {z}+\ vec {q}}β，\ vec，\ vec {q}} $β= 2 $吸引子，$s_β$，是一种称为sierpiński垫圈（或筛子）的分形，对于$β> 2 $，它也是分形。我们的目标是研究吸引子上的贪婪，懒惰和随机的$β$转换，以$ 1 <β<2 $。对于$ 1 <β\ leq 3/2 $，$s_β$是一个三角形，这表明贪婪的转换$t_β$和懒惰的转换$l_β$是同构的，它们都承认了绝对连续的不变措施。我们表明，$s_β$中$ \ vec {z} $的所有$β$ - 可以通过$ \ {0,1 \}^\ mathbb {n} n} \ times \ times \ times \ {0,1,1,2 \}^\ m mathbb {n} s s n} s n} s n} s n} s n} s n}的随机地图$k_β$可以生成$s_β$ in $s_β$ in $s_β$。当$ 1 <β\leqβ_*$时，最大熵的最大熵，其中$β_*\大约1.4656 $是$ x^3-x^2-1 = 0 $的根。我们还显示了$k_β$ - invariant概率度量的存在，与$ m_1 \ otimes m_2 \ otimesλ_2$相对于$ \ {0,1 \}^\ natbb {n}，$ mathbb {n}，$ \ n}，$ m_1，$ m_2 $，是$ m_2 \ otimesλ_2$λ_2$λ_2$ $λ_2$是$s_β$的标准化lebesgue度量。对于$ 3/2 <β\leqβ^*$，其中$β^*\约1.5437 $是$ x^3-2x^2+2x = 2 $的根，$s_β$中有径向孔。在这种情况下，$k_β$是在$ \ {0,1 \}^\ mathbb {n} \ timess_β$上定义的。我们还表明它具有最大熵的独特不变度量。
We consider the iterated function system (IFS) $$f_{\vec{q}}(\vec{z})=\frac{\vec{z}+\vec{q}}β,\vec{q}\in\{(0,0),(1,0),(0,1)\}.$$ As is well known, for $β= 2$ the attractor, $S_β$, is a fractal called the Sierpiński gasket(or sieve) and for $β>2$ it is also a fractal. Our goal is to study greedy, lazy and random $β$-transformations on the attractor for this IFS with $1<β<2$. For $1<β\leq 3/2$, $S_β$ is a triangle and it is shown that the greedy transformation $T_β$ and the lazy transformation $L_β$ are isomorphic and they both admit an absolutely continuous invariant measure. We show that all $β$-expansions of a point $\vec{z}$ in $S_β$ can be generated by a random map $K_β$ defined on $\{0,1\}^\mathbb{N}\times\{0,1,2\}^\mathbb{N}\times S_β$ and $K_β$ has a unique invariant measure of maximal entropy when $1<β\leqβ_*$, where $β_*\approx 1.4656$ is the root of $x^3-x^2-1=0$. We also show existence of a $K_β$-invariant probability measure, absolutely continuous with respect to $m_1\otimes m_2 \otimes λ_2$, where $m_1, m_2$ are product measures on $\{0,1\}^\mathbb{N},\{0,1,2\}^\mathbb{N}$, respectively, and $λ_2$ is the normalized Lebesgue measure on $S_β$. For $3/2<β\leq β^*$, where $β^*\approx 1.5437$ is the root of $x^3-2x^2+2x=2$, there are radial holes in $S_β$. In this case, $K_β$ is defined on $\{0,1\}^\mathbb{N}\times S_β$. We also show that it has a unique invariant measure of maximal entropy.
