We prove that for any integer $kgeq 2$ and $varepsilon>0$, there is an integer $ell_0geq 1$ 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 $ellgeq ell_0$ 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, Kuhn 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.
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