Negative Lyapunov exponent of circle maps forced by expanding circle endomorphisms

We study maps on the torus $\mathbb{T}^2$ that are of the form $F(x,y) = (bx, f_x(y))$, where $b\geq 2$ is an integer. We establish an open class of $C^1$-maps, with $f_x(y)$ that are typically non-monotonic in $x$, for which the Lyapunov exponents on the fibre are negative almost everywhere. For each fixed $f_x(y)$ and a base map $bx$ that is sufficiently expanding, we establish a uniform upper bound for the Lyapunov exponents; moreover, the uniform bound depends on selective characteristics of $f$. This implies that orbits on the same fibre exhibit local synchronisation.