Given any numeration system, we call carry propagation at a number $N$ the number of digits that are changed when going from the representation of $N$ to the one of $N+1$, and amortized carry propagation the limit of the mean of the carry propagations at the first $N$ integers, when $N$ tends to infinity, if this limit exists. In the case of the usual base $p$ numeration system, it can be shown that the limit indeed exists and is equal to $p/(p-1)$. We recover a similar value for those numeration systems we consider and for which the limit exists. We address the problem of the existence of the amortized carry propagation in non-standard numeration systems of various kinds: abstract numeration systems, rational base numeration systems, greedy numeration systems and beta-numeration. We tackle the problem by three different types of techniques: combinatorial, algebraic, and ergodic. For each kind of numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.

中文翻译：

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后继函数的进位传播

给定任何计数系统，我们将数字 $N$ 处的进位传播称为从 $N$ 的表示变为 $N+1$ 之一时更改的位数，摊销进位传播称为均值的极限如果存在此限制，则当 $N$ 趋于无穷大时，前 $N$ 个整数处的进位传播的数量。在通常的基础 $p$ 计算系统的情况下，可以证明极限确实存在并且等于 $p/(p-1)$。我们为我们考虑的那些存在限制的计数系统恢复了类似的值。我们解决了在各种非标准计算系统中存在摊销进位传播的问题：抽象计算系统、有理基础计算系统、贪婪计算系统和 beta 计算系统。我们通过三种不同类型的技术来解决这个问题：组合、代数和遍历。对于我们考虑的每种计数系统，相关方法都允许为进位传播的存在建立充分条件，并且示例表明这些条件接近必要条件。