A New Approach To Find The Invariant Densities Of Piecewise Linear Maps Via δ Function

Invariant densities is a core area in the research of dynamical system. This thesis focuses on finding the invariant densities of piecewise linear maps on the intervalI= [0,1].Finding the fixed points of Frobenius-Perron problem:Pτf=f, is the common way to get invariant densities. During the process of solving this problem, this thesis introduces δ function by adopting the similar ideas of generalized derivative and solving fundamental solution for linear PDEs. And then assume the invariant densities with a specific form by virtue of conclusions from other researchers. Finally, the Frobenius-Perron equations are reduced to series, and we are able to find the invariant densities of piecewise linear maps via the properties of orbits of endpoints.This thesis discusses the invariant densities of four different piecewise linear maps in chapter three, they are Lorenz map with τ(0)= 0, the map of raising and falling, full Lorenz map and compressive map respectively. In particular, we could find the solution in the first three maps, but the last one has no solution, which corresponds with non-existence of its invariant density.