The Sub-additive Thermodynamic Formalism And The Dimension Estimates For Random Hyperbolic Set

The topological pressure plays a fundamental role in ergodic theory,dynamical systems,dimension theory and the theory of equilibrium states.For classical thermo-dynamic formalism,i.e.,the potentials is additive,both in deterministic and random case,there are a great deal of classical results.The Hausdorff dimension and related properties of conformal repeller can be well described by the classical thermodynamic formalism,but it can not work well in non-conformal dynamic system.So the nonadditive thermodynamic formalism arise the expert's interest,and they have already obtained some exciting results.Particularly,Barreira,Falconer and Cao,feng,huang developed different versions of non-additive thermodynamic formalism respectively,and they found that it has nice applications in dimension theory respectively.Their works said that the dimension of non-conformal repeller can be well estimated by the root of different version of non-additive topological pressure.On the other hand,some experts was dedicated to find the relation between dimension and the system's invariants,such as entropy and Lyapunov exponents,and they also obtained some wonderful results. In this paper,we studied sub-additive thermodynamic formalism and the dimension estimates in random dynamical systems.In chapter one,we briefly recalled the history of thermodynamic formalism both in deterministic and random case.At the same time,we simply introduced the content in dimension theory,in particularly,we emphasized the application of thermodynamic formalism in dimension theory.Besides,we introduced some recent results related to thermodynamic formalism and dimension theory.In chapter two,applying the sub-additive thermodynamic formalism developed by Cao,Feng and Huang,we introduced a definition of sub-additive measure-theoretic pressure by using spanning set.While we introduced another definition of sub-additive measure-theoretic pressure by using the theory of caratheodory dimension characteristic, and we found that this measure-theoretic pressure was equal to the topological pressure on a certain set.Furthermore,we proved that this two definitions was equivalent under some assumptions. In chapter three,under Yuri Kifer's random set up,we generalized Cao,Feng and Huang's results to random case.We also defined the topological pressure with an arbitrary family of random functions,and we gave some properties and an application of the new defined topological pressure.In chapter four,we considered the dimension estimates of random non-conformal repellers.And we showed that the dimension estimates can be given by random topological entropy,topological pressure and uniform Lyapunov exponents.A measure-theoretic version was also contained.In chapter five,we introduced a random dynamical systems which was generated by forward and backward successive applications of randomly chosen maps from the space formed by all C~2 diffeomorphisms on some manifold,these maps being independent and identically distributed with a certain law.Under this system,we gave a formula which related entropy,dimension and Lyapunov exponents in two dimension.In the appendix,we gave a proof of the random version of Brin-Katok theorem.