The Variational Principle Of Packing Entropy Dimension

In this paper,we define the ?-topological entropy and the topological entropy di-mension,based on topological entropy,and study their basic properties,give and prove the corresponding variational principle.The specific content is as follows:Firstly,we define the ?-local upper entropy by using ideas similar to Brin-Katok,define the open covering ?-topological entropy by using a method similar to the definition of open covering entropy,define the upper capacity ?-topological entropy by using a method similar to the definition of upper capacity topological entropy,and define the Packing ?-topological entropy htapP(T,Z,?)of any subset Z by using a method similar to the definition of Packing topological entropy.Then,we give the product formulas of the upper capacity ?-topological entropy and the Packing ?-topological entropy:Secondly,we draw on the fractal dimension and use the critical property of ?-entropy to define the Packing topological entropy dimension DP(T,Z)of any subset Z and the lo-cal upper entropy dimension(?)about the given measure ?.Then we give and prove the variational principle of Packing ?-topological entropy and the variational principle of Packing topological entropy dimension:Finally,We give some applications of ?-topological entropy,entropy dimension,and variational principles.Through the example 1,we explained the reason why the vari-ational principle chooses Packing topological entropy dimension,but not the capacity topological entropy dimension.Through example 2,we give the proof of the classic Garden theorem,and extend the classic Garden theorem to higher dimensions.