  We show that part I of uniform Martin's conjecture follows from a local
phenomenon, namely that if a non-constant Turing invariant function goes from
the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then
$\boldsymbol x \le_T \boldsymbol y$. Besides improving our knowledge about part
I of uniform Martin's conjecture (which turns out to be equivalent to Turing
determinacy), the discovery of such local phenomenon also leads to new results
that did not look strictly related to Martin's conjecture before. In
particular, we get that computable reducibility $\le_c$ on equivalence
relations on $\mathbb N$ has a very complicated structure, as $\le_T$ is Borel
reducible to it. We conclude raising the question "Is part II of uniform
Martin's conjecture implied by local phenomena, too?" and briefly indicating a
possible direction.
