Boundary influence on the entropy of a Lozi-type map
The paper deals with \(n\)th-degree Lozi-type map \[ T(x,y) = (y, F(y) - bx), \] where \(F(y)\) is a continuous piece-wise linear function. This map is induced from a spatial discretization of a reaction-diffusion system. The authors solve open problems from the work by \textit{V. S. Afraimovich} and \textit{S. B. Hsu} [Lectures on chaotic dynamical systems. AMS/IP Studies in Advanced Mathematics 28. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2003; Zbl 1152.37300)] and prove that the equalities \[ h(T) = h_D(T)= h_N(T) = h_{l(1),l(2)}(T) \] are not true in general. Here \(h(T)\) and \(h_{l(1),l(2)}(T)\) are the so-called spatial entropies of the map \(T\) and the spatial entropies of \(T\) with respect to the lines \(l(1)\) and \(l(2),\) and \(h_D(T)\) and \(h_N(T)\) are the spatial entropies with respect to the Dirichlet and Neuman boundary conditions. Sufficient conditions under which the equalities hold are given.
scientific article; zbMATH DE number 5777347