Geodesic
Energy function for direct products of discrete dynamical systems
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 2, pp. 11-21.

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This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.
Keywords: direct product, diffeomorphism, chain recurrent set, energy function 
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M. K. Barinova; E. K. Shustova. Energy function for direct products of discrete dynamical systems. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 2, pp. 11-21. http://geodesic.mathdoc.fr/item/SVMO_2023_25_2_a0/

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