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We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive $S^1$-equivariant symplectic homology can be computed; it is generated by those orbits. We prove a "Viterbo functoriality" property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive $S^1$-equivariant symplectic homologies and morphisms compose nicely. These properties allow us to give a proof of Ustilovsky's result on the number of non isomorphic contact structures on the spheres $S^{4m+1}$. They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in $\mathbb{R}^{2n}$. We extend this result to some hypersurfaces in some negative line bundles.

The positive equivariant symplectic homology as an invariant for some contact manifolds