We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associ...We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.展开更多
基金This work was supported by National Science Foundation of USA(Grant No.DMS-1747905)the Simons Foundation(Grant No.523341)+1 种基金Professional Sta Congress of the City University of New York Enhanced Award(Grant No.62777-0050)National Natural Science Foundation of China(Grant No.11571122).
文摘We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials.We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials.We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures.We prove that with this complex Banach manifold structure,the space is complete and,moreover,is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures.There is a maximum metric on the space,which is incomplete.We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same.We prove that a geometric Gibbs measure is an equilibrium state,and the in mum of the metric entropy function on the space is zero.
