Balkan Mathematical Olympiad

Yes, it is true. Suppose that $f$ is $k$-jumpy. A number of the form $z+\frac{1}{2}$ where $z$ is an integer is called fence. Select a fence. Count the number of integers which $f$ causes to jump from left to right over the fence, minus the number of integers it causes to jump from right to left over the same fence. Since $f$ is $k$-jumpy, the integers being subtracted are both in the range $0$ to $k$, so their difference has modulus at most $k$. We call this quantity, jumps to the right minus jumps to the left, the flow over this fence.

Choose any two fences with distance at least $k$ apart. By the Dirichlet principle, the flows over the two fences must be the same. Now by varying the locations of these fences, we see that the flows over all fences are the same, so it makes sense to talk about the flow of the map $f$, independent of the fence.

Let $t:\mathbb{Z}\to \mathbb{Z}$ be defined by $x\mapsto x+1$ for each integer $x$, a 1-jumpy bijection of flow $1$. By precomposing $f$ with $g$, a suitable power of $t$ or $t^{-1}$, we obtain a $k_1$-jumpy bijection $g\circ f$ of flow $0$. Notice that $g$ (and therefore $g^{-1}$) is a composition of 1-jumpy bijections.

Next we select a (bi)-infinite arithmetic progression of fences with common difference which is large compared with $k_1$. Choose the common difference $10k_1$ say. Now there are at most $2k_1$ integers which $g\circ f$ causes to jump across a given fence, the same number in each direction. For the fence $h$ in the AP, let $X_h$ denote the $2k_1$ integers which have distance at most $k_1$ from $h$. Let $X$ denote the union of the sets $X_h$ as $h$ runs over the fences in the AP. We choose a bijection $l$ which restricts to a bijection from $X_h$ to $X_h$ for each $h$ in the AP, and is the identity map outside $X$. On $X_h$, $l$ returns whence they came those elements which $g\circ f$ caused to jump the fence $h$, and keeps any other elements of $X_h$ on the same side of the fence where they are found. Such a map $l$ is $k_1$-jumpy, so $l\circ g\circ f$ is $2k_1$-jumpy, has zero flow, and causes no integer to jump a fence $h$ in the AP.

Therefore $l\circ g\circ f$ has cycles of length at most $10k_1$. Any finitary permutation is product of transpositions of elements in its support (the number of them being bounded by a function of the size of the support), and any transposition of elements at most $10k_1$ apart is the product of $20k_1-1$ adjacent transpositions all with support in the (closed) interval defined by the two elements being transposed, for example: $$(1,4)=(1,2)(2,3)(3,4)(2,3)(1,2).$$ Permutations with cycle structure consisting of adjacent transpositions are 1-jumpy.

Therefore we can express $l\circ g\circ f$ as a product of 1-jumpy bijections, where we work on each run of integers between fences separately and in parallel (using the identity map in consecutive ranges where appropriate). In similar fashion, we can express $l$ as a product of 1-jumpy bijections.

Precomposing with $g^{-1}\circ l^{-1}$, we express $f$ as a product of 1-jumpy bijections, as required.