  Given a dynamic graph $G$ with $n$ vertices and $m$ edges subject to
insertion an deletions of edges, we show how to maintain a
$(1+\varepsilon)\Delta$-edge-colouring of $G$ without the use of randomisation.
  More specifically, we show a deterministic dynamic algorithm with an
amortised update time of $2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$
using $(1+\varepsilon)\Delta$ colours. If $\varepsilon^{-1} \in
2^{O(\log^{0.49} n)}$, then our update time is sub-polynomial in $n$.
  While there exists randomised algorithms maintaining colourings with the same
number of colours [Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya,
Costa, Panski, Solomon SODA'24] in polylogarithmic and even constant update
time, this is the first deterministic algorithm to go below the greedy
threshold of $2\Delta-1$ colours for all input graphs.
  On the way to our main result, we show how to dynamically maintain a shallow
hierarchy of degree-splitters with both recourse and update time in $n^{o(1)}$.
We believe that this algorithm might be of independent interest.
