Directional Topological Pressure And Directional Measure For Z_+~k-actions

A classical discrete dynamical system can be regarded as a Z-action.However,higher rank action,such as Z~k-action,is more complicated,which makes it difficult to be investigated.Using creative techniques“Coding”and“Shading”,the definition of entropy and basic properties of ex-pansion subdynamics of Z~k-actions were given by M.Boyle and D.Lind in 1997.Furthermore,some perfect results were obtained,such as the continuity of directional entropy within an expan-sive component and a variantional principle.The main aim of this thesis is to give the concept of directional topological pressure and directional measure for Z_+~k-actions.In the first part,we get one dimensional subdynamics of Z_+~k-actions induced by unit direc-tional vector,after that we give the definition of directional topological pressure of this subdy-namics by dint of non-autonomous dynamics theory and prove the rationality of this definition and some basic properties.In the second part,the above one dimensional subdynamics is transformed into a one-sided canonical i.i.d.random dynamical system.Applying the random dynamical system theorey,we obtain the variantional principle inequality of directional topological pressure.In the third part,in order to research invariant measure of non-autonomous dynamics,we consider positively expanding Z_+~k-actions of class C~2and give the concept of directional measure of them.Moreover,we show that above directional measure is absolutely continuous with respect to a Lebesgue measure and it is continuous with respect to the direction.