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We prove a contact non-squeezing result for a class of embeddings betweenstarshaped domains in the contactization of the symplectization of the unitcotangent bundle of certain manifolds. The class of embeddings includesembeddings which are not isotopic to the identity. This yields a new proof thatthere is no positive loop of contactomorphisms in the unit cotangent bundlesunder consideration. The proof uses the shape invariant introduced by Sikoravand Eliashberg.

Contact non-squeezing and orderability via the shape invariant