Entropy And Pressure Along Unstable Foliations For Smooth Random Dynamical Systems

In dynamical systems,entropy is an important invariant,which describes the com-plexity of systems.Topological entropy and metric entropy give the complexity of orbits from geometric and statistical points of view respectively.The famous variational prin-ciple develops the inherent relationship between them.In the research of differentiable dynamical systems,random dynamical systems and ergodic theory,the famous entropy formula shows that positive metric entropy originates from positive Lyapunov exponents.A natural question is that how to define metric entropy and topological entropy reasonably and obtain the variational principle when the problem is concentrated on the unstable manifolds corresponding to positive Lyapunov exponents?Recently,some researchers investigated this question for partially hyperbolic diffeomorphisms systematically,intro-duced unstable metric entropy,unstable topological entropy and unstable pressure,gave the corresponding variational principles.The main purpose of this paper is to study the unstable entropies and unstable pressure for random dynamical systems and endomorphisms.Especially various versions of variational principles are given.There are two parts in this paper:In the first part(Chapter 1 and Chapter 2)we consider the unstable entropies and unstable pressure for random case.Let F be a C~2partially hyperbolic random dynamical system.In Chapter 1,for the unstable foliation,the corresponding unstable metric entropy,unstable topological entropy and unstable pressure via the dynamics of F on the unsta-ble foliation are introduced and investigated.A version of Shannon-McMillan-Breiman Theorem for unstable metric entropy is given,and a variational principle for unstable pressure(and hence for unstable entropy)is obtained.Moreover,as an application of the variational principle,equilibrium states for the unstable pressure including Gibbs u-states are investigated.In Chapter 2,for unstable foliation,the corresponding local unstable metric entropy,local unstable topological entropy and local unstable pressure via the dynamics of F are introduced and investigated.And variational principles for local unstable entropy and local unstable pressure are obtained respectively.In the second part(Chapter 3 and Chapter 4),we consider the unstable entropies and unstable pressure for endomorphisms.In Chapter 3,unstable metric entropy,unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.A ver-sion of Shannon-McMillan-Breiman Theorem is established,and a variational principle is formulated,which gives a relationship between unstable metric entropy and unstable pressure(unstable topological entropy).As an application of the variational principle,some results on the u-equilibrium states are given.In Chapter 4,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and inves-tigated.Specially,two variational principles concerning relationships among the above mentioned numbers are formulated.