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Let $n$ be a positive integer. An $n$-cycle of linear mappings is an$n$-tuple $(u_1,\dots,u_n)$ of linear maps $u_1 \in \mathrm{Hom}(U_1,U_2),u_2\in \mathrm{Hom}(U_2,U_3),\dots,u_n \in \mathrm{Hom}(U_n,U_1)$, where$U_1,\dots,U_n$ are vector spaces over a field. We classify such cycles, up toequivalence, when the spaces $U_1,\dots,U_n$ have countable dimension and thecomposite $u_n\circ u_{n-1}\circ \cdots \circ u_1$ is locally finite.  When $n=1$, this problem amounts to classifying the reduced locally nilpotentendomorphisms of a countable-dimensional vector space up to similarity, and theknown solution involves the so-called Kaplansky invariants of $u$. Here, weextend Kaplansky's results to cycles of arbitrary length. As an application, weprove that if $u_n \circ \cdots \circ u_1$ is locally nilpotent and the $U_i$spaces have countable dimension, then there are bases$\mathbf{B}_1,\dots,\mathbf{B}_n$ of $U_1,\dots,U_n$, respectively, such that,for every $i \in \{1,\dots,n\}$, $u_i$ maps every vector of $\mathbf{B}_i$either to a vector of $\mathbf{B}_{i+1}$ or to the zero vector of $U_{i+1}$(where we convene that $U_{n+1}=U_1$ and $\mathbf{B}_{n+1}=\mathbf{B}_1$).

Locally finite cycles of linear mappings in countable dimension