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Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of “chain graph” on “epsilon-chains”, defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every “chain graph” of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses. 

The tent map family is arguably the simplest one-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works, the second author, jointly with J. Yorke, studied the structural graph and backward limits of S-unimodal maps. In this article, we generalize those results to tent-like unimodal maps. By tent-like here we mean maps that share fundamental properties that characterize tent maps, namely unimodal maps without wandering intervals nor attracting periodic orbits and whose structural graph has a finite number of nodes. This article was started under grant DMS-1832126 and then completed and published under grant DMS-2308225. 

While studying gradient dynamical systems, Morse introduced the idea of encoding the qualitative behavior of a dynamical system into a graph. Smale later refined Morse’s idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale’s vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node M to node N (we say that N is downstream from M) if the unstable manifold of M intersects the stable manifold of N. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale’s construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set, introduced by Birkhoff in 1920s, were elaborated first by Auslander in early 1960s, by Conley in early 1970s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the prolongational relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations (which we call streams) containing the space of orbits of a discrete-time or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of dynamical systems. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. Our main general result is that each stream of a semi-flow with “compact dynamics” has a connected graph. The range of semi-flows covered by our theorem goes from 1-dimensional discrete-time systems like the logistic map up to infinite-dimensional continuous-time systems like the semi-flow of quasilinear parabolic reaction–diffusion partial differential equations.