  We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an
integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices
with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional
decomposition into tight cycles of length $\ell$ ($\ell$-cycles for short)
whenever $\ell\geq \ell_0$ and $n$ is large in terms of $\ell$. This is
essentially tight.
  This immediately yields also approximate integral decompositions for these
hypergraphs into $\ell$-cycles. Moreover, for graphs this even guarantees
integral decompositions into $\ell$-cycles and solves a problem posed by Glock,
K\"uhn and Osthus. For our proof, we introduce a new method for finding a set
of $\ell$-cycles such that every edge is contained in roughly the same number
of $\ell$-cycles from this set by exploiting that certain Markov chains are
rapidly mixing.
