Formula Of Entropy Along Unstable Foliations For C~1 Diffeomorphisms With Dominated Splitting

In this thesis,the relation between the entropy along unstable foliations for C1 diffeomorphisms with dominated splitting and Lyapunov exponents is considered,showing that the contribution of Lyapunov exponents in different levels to the corresponding entropy under the condition of "C1 + dominated splitting".This thesis includes two parts.In part one,for a diffeomorphism f with dominated splitting on a compact Riemannian manifold M without boundary,we give an estimation of hμi(f)the entropy along the ith unstable foliation Wi from above for an f-invariant measure μ.Specifically,where λ1(x)>λ2(x)>…>λu(x)(x)are the positive Lyapunov exponents at x,mj(x)is the multiplicity of λj(x),u(i,x)=u(x)-i+1,and Γi is the set of x such that u(i,x)>0.In part two,for some invariant measure μ with some absolute continuity,we give the estimation of hμu(f)from below,which implies the entropy formula.Specifically,if for μ-a.e.x∈Γi and every measurable partition ζi subordinate to Wi we have (?),where {μxζi} is a canonical system of conditional measures and λxi is the corresponding Riemannian measure on Wi(x),the following formula is given.