论文标题
拓扑熵和因子复杂性在单体形态和自由组基础下的行为如何变化？
How do topological entropy and factor complexity behave under monoid morphisms and free group basis changes ?
论文作者
论文摘要
对于任何不播种的免费单型形态$σ：\ cal a^* \ to \ cal b^* $，对于任何shift $ shift $ x \ subset \ cal a^\ z $及其图像subshift $ y =σ（x）\ subset \ subset \ cal b^\ z $ 0 $ $ c \ cdot p_x（d \ cdot n）\，\，\ leq \，\，\，p_y（n）\，\，\，\，\ leq \，\，c \ cdot p_x（n）$ $可容纳所有足够大的大整数$ n $，前提是$ n $ in $ n $，只要$ n $ nes $ nes $ nes $ iniz in $ nes $ iniz in $ nes $ in。如果$σ$是另外的字母到字母，则$ p_y $属于$θ（p_x）$（相反）。但是，否则，在某些示例中，$ p_x $不在$ \ cal o（p_y）$中。 因此，通常，即使$σ$在$ x $中可识别，将$ x $的拓扑熵的$ h_x $ $ h_x $都无法保留。 结果，没有有意义的方法来定义自由组$ f_n $上电流的拓扑熵；只有电流的区别$μ$与拓扑熵的区别$ h _ {\ tiny \ supp（μ）} = 0 $和$ h _ {\ tiny \ supp（μ）}> 0 $。
For any non-erasing free monoid morphism $σ: \cal A^* \to \cal B^*$, and for any subshift $X \subset \cal A^\Z$ and its image subshift $Y = σ(X) \subset \cal B^\Z$, the associated complexity functions $p_X$ and $p_Y$ are shown to satisfy: there exist constants $c, d, C > 0$ such that $$c \cdot p_X(d \cdot n) \,\, \leq \,\, p_Y(n) \,\, \leq \,\, C \cdot p_X(n)$$ holds for all sufficiently large integers $n \in \N$, provided that $σ$ is recognizable in $X$. If $σ$ is in addition letter-to-letter, then $p_Y$ belongs to $Θ(p_X)$ (and conversely). Otherwise, however, there are examples where $p_X$ is not in $\cal O(p_Y)$. It follows that in general the value $h_X$ of the topological entropy of $X$ is not preserved when applying a morphism $σ$ to $X$, even if $σ$ is recognizable in $X$. As a consequence, there is no meaningful way to define the topological entropy of a current on a free group $F_N$; only the distinction of currents $μ$ with topological entropy $h_{\tiny\supp(μ)} = 0$ and $h_{\tiny\supp(μ)} > 0$ is well defined.
