Parametrized Euler class and semicohomology theory

We extend Ghys' theory about semiconjugacy to the world of measurable cocycles. More precisely, given a measurable cocycle with values into $\text{Homeo}^+(\mathbb{S}^1)$, we can construct a $\text{L}^\infty$-parametrized Euler class in bounded cohomology. We show that such a class vanishes if and only if the cocycle can be lifted to $\text{Homeo}^+_{\mathbb{Z}}(\mathbb{R})$ and it admits an equivariant family of points. We define the notion of semicohomologous cocycles and we show that two measurable cocycles are semicohomologous if and only if they induce the same parametrized Euler class. Since for minimal cocycles, semicohomology boils down to cohomology, the parametrized Euler class is constant for minimal cohomologous cocycles. We conclude by studying the vanishing of the real parametrized Euler class and we obtain some results of elementarity.