  We prove a new concentration inequality for U-statistics of order two for
uniformly ergodic Markov chains. Working with bounded and $\pi$-canonical
kernels, we show that we can recover the convergence rate of Arcones and
Gin{\'e} who proved a concentration result for U-statistics of independent
random variables and canonical kernels. Our result allows for a dependence of
the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of
standard blocking tools. Our proof relies on an inductive analysis where we use
martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's
type inequality. Assuming further that the Markov chain starts from its
invariant distribution, we prove a Bernstein-type concentration inequality that
provides sharper convergence rate for small variance terms.
