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 ) = h D ( T ) = h N ( T ) = h ℓ ( 1 ) , ℓ ( 2 ) ( T ) ? Here h ( T ) and h ℓ ( 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 h D ( T ) and h N ( 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.