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The main result of this paper is the existence of a hyperinvariant subspaceof weighted composition operator $Tf=vf\circ\tau$ on $L^p([0,1]^d)$, ($1 \leq p\leq \infty$) when the weight $v$ is in the class of ``generalizedpolynomials'' and the composition map is a bijective ergodic transformsatisfying a given discrepancy. The work is based on the construction of afunctional calculus initiated by Wermer and generalized by Davie.

Hyperinvariant subspace for weighted composition operator on $L^p([0,1]^d)$