Phase transition for piecewise linear fibonacci bimodal map

In this paper we concern with the Fibonacci bimodal maps. We first study the topological properties of the Fibonacci bimodal maps in the context of kneading map and give an equivalent description of Fibonacci combinatorics. Then we construct a one-parameter family $f_{\lambda }$ of countably piecewise linear Fibonacci bimodal maps depending on the parameter $\lambda$ which are all odd functions. By a random walk argument on its induced Markov map, we will show that a phase transition occurs from Lebesgue conservative to Lebesgue dissipative behaviors.