  We address the problem of estimating the mixing time of a Markov chain from a
single trajectory of observations. Unlike most previous works which employed
Hilbert space methods to estimate spectral gaps, we opt for an approach based
on contraction with respect to total variation. Specifically, we estimate the
contraction coefficient introduced in Wolfer [2020], inspired from Dobrushin's.
This quantity, unlike the spectral gap, controls the mixing time up to strong
universal constants and remains applicable to non-reversible chains. We improve
existing fully data-dependent confidence intervals around this contraction
coefficient, which are both easier to compute and thinner than spectral
counterparts. Furthermore, we introduce a novel analysis beyond the worst-case
scenario by leveraging additional information about the transition matrix. This
allows us to derive instance-dependent rates for estimating the matrix with
respect to the induced uniform norm, and some of its mixing properties.
