Boundary influence on the entropy of a Lozi-type map

Let T be a Henon-type map induced from a spatial discretization of a reaction–diffusion system. With the above-mentioned description of T, the following open problems were raised in [V.S. Afraimovich, S.B. Hsu, Lectures on Chaotic Dynamical Systems, AMS International Press, 2003]. Is it true that, in general, h(T)=hD(T)=hN(T)=hℓ(1),ℓ(2)(T)? Here h(T) and hℓ(1),ℓ(2)(T) (see Definitions 1.1 and 1.2) are, respectively, the spatial entropy of the system T and the spatial entropy of T with respect to the lines ℓ(1) and ℓ(2), and hD(T) and hN(T) are spatial entropy with respect to the Dirichlet and Neuman boundary conditions. If it is not true, then which parameters of the lines ℓ(i), i=1,2, are responsible for the value of h(T)? What kind of bifurcations occurs if the lines ℓ(i) move? In this paper, we show that this is in general not always true. Among other things, we further give conditions for which the above problem holds true.