arXiv:0905.1308v2 Announce Type: replace Abstract: Let $\gamma:[0,1]\to \mathbb{R}^2$ be a continuous curve with $\gamma(0)=(0,0)$ and $\gamma(1)=(1,1)$. We prove that if $\gamma(t)\in(0,1)^2$ for all $t\in(0,1)$, then for each $n\in \mathbb{N}$ there exist points $A_1,\dots,A_n$ on $\gamma$ such that, with $A_0=(0,0)$ and $A_{n+1}=(1,1)$, the horizontal increments $\pi_1(\overrightarrow{A_iA_{i+1}})$ and the vertical increments $\pi_2(\overrightarrow{A_iA_{i+1}})$, $0\le i\le n$, are positive and coincide up to order. More generally, the same conclusion holds as soon as $\gamma$ contains an arc that joins two points of the diagonal segment, lies on one side of the diagonal, and satisfies a mild one-sided boundary condition. This covers, for instance, the graph of every continuous function $f$ with $f(0)=0$ and $f(1)=1$, with no restriction on the range of $f$. In all cases the rearrangement can be taken to be a cyclic shift by one place of a block of consecutive increments. The main tool is a version of the mountain climbers' theorem in which one function is piecewise monotone and the other is an arbitrary continuous function. As an application, any two non-atomic probability measures on $[0,1]$ satisfying a mild condition admit, for each $n$, a partition of $[0,1]$ into $n+1$ consecutive intervals whose two lists of measures are rearrangements of each other.
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