Fractional cycle decompositions in hypergraphs

We prove that for any integer k >= 2 and epsilon>0, there is an integer l0 >= 1 such that any k-uniform hypergraph on n vertices with minimum codegree at least (1/2+epsilon)n has a fractional decomposition into (tight) cycles of length l (l-cycles for short) whenever l >= l0 and n is large in terms of l. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into l-cycles. Moreover, for graphs this even guarantees integral decompositions into l-cycles and solves a problem posed by Glock, Kuhn, and Osthus. For our proof, we introduce a new method for finding a set of l-cycles such that every edge is contained in roughly the same number of l-cycles from this set by exploiting that certain Markov chains are rapidly mixing.