We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

中文翻译：

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平衡的强移位等价，平衡的 in-splits 和最终的共轭

我们引入了平方非负整数矩阵之间平衡强移位等价的概念，并表明当且仅当它们的邻接矩阵与平衡强移位等价矩阵共轭时，两个没有汇的有限图最终是单边共轭的。此外，我们证明了这些图最终是共轭的，当且仅当一个可以被另一个通过一系列 out-splits 和平衡的 in-splits 到达，后者是 Williams 引入的经典 in-split 移动的变体在他对有限类型移位的研究中。我们还将单边最终共轭与图的有限路径上的某些块映射相关联。这些特征强调最终的共轭是双边共轭的单面模拟。