On The Variational Principle For A Class Of Skew Product System

In this thesis,by using fiber specification property,we establish a variational principle between the fiber topological entropy on conditional level sets of Birkhoff average and fiber measure-theoretical entropy for the skew product system driven by uniquely ergodic homeomorphism system satisfying Anosov and topological mixing on fiber property.The thesis is divided into five chapters and organized as follows:In chapter 1,we clarify the research background and motivation of this thesis.Based on some basic definitions given in section 1.2,1.3,1.4,we state our main results in section 1.5.In chapter 2,we give some examples of skew product system driven by an uniquely ergodic homeomorphism satisfying Anosov and topological mixing on fiber property.In chapter 3,we give some preliminary knowledge and lemmas for the proof of our main results.The hyperbolic dynamics of the target systems are given in section 3.1.In section 3.2,we compare our definition of fiber specification property with the specification property proposed by Gundlach and Kifer in(17).The basic concept of fiber measure-theoretical entropy,a measure approximation lemma and random version of Katok’s entropy formula are given in section 3.3 and section 3.4.Finally,we give some knowledge related to entropy maps and equilibrium states in section 3.5.In chapter 4,we prove our main results.In section 4.1,we establish the fiber specification property for our target systems.In section 4.2,we establish a variational principle between the fiber topological entropy on conditional level sets of Birkhoff average and fiber measure-theoretical entropy utilizing an intermediate quantity Λφ,α(ω).To achieve this goal,we prove a crucial lemma 4.2 and construct fiber Moran-like fractals based on Takens and Verbitskiy’s idea of constructing Moran fractal(35).The theorem 2 will finally be proved using entropy distribution principle type argument.In section 4.3,we establish the relationship between the fiber topological entropy on conditional level sets of Birkhoff average of a continuous observable φ and the Legendre transform of its fiber topological pressure function.In section 4.3,we establish a similar variational principle for the local entropies of every H?lder continuous potential.In chapter 5,we collect the detailed proof of all lemmas in previous sections.