Abstract
Following the ideas of A. Lyapunov C. Conley introduced the notion of a Lyapunov function for a dynamical system (see Definition 7.1). In 1978 he proved the existence of a continuous Lyapunov function for every dynamical system [2]. This result is called the fundamental theorem of dynamical systems. If a Lyapunov function is smooth and the set of its critical points coincides with the chain recurrent set then this function is called the energy function. Very generally smooth flows admit an energy function (see, e.g. Theorem 6.12 in [1]), but it is not true for diffeomorphisms. First results on construction of an energy function (see Definition 7.2) belong to S. Smale. In 1961 [9] he proved the existence of an energy function, which is a Morse function, for every gradient-like flow (i.e. Morse-Smale flow without closed trajectories). K. Meyer [7] in 1968 generalized this result and constructed an energy function, which is a Morse-Bott function, for an arbitrary Morse-Smale flow. The only result of this kind for diffeomorphisms belongs to D. Pixton [8], who in 1977 proved the existence of an energy function, which is a Morse function, for Morse-Smale diffeomorphisms on surfaces. Furthermore, he constructed a diffeomorphism on the 3-sphere (we have already mentioned it in Chapter 4 as the Pixton’s example) which has no energy function, and he explained the phenomenon to be caused by the wild embedding of the separatrices of the saddle points. Recently the conditions of existence of an energy function were found in [3–6]. In section 7.1 we present important properties of a Lyapunov function, which is a Morse function, for Morse-Smale diffeomorphisms on n-manifolds. In section 7.2 we introduce a dynamically ordered Morse-Lyapunov function for an arbitrary Morse-Smale diffeomorphism of a 3-manifold with the properties closely related to the dynamics of the diffeomorphism. We show that the necessary and sufficient conditions of the existence of an energy function with these properties are determined by the type of the embedding of the 1-dimensional attractors (repellers), each of which is the union of the 0-dimensional and the 1-dimensional unstable (stable) manifolds of the periodic points of the diffeomorphism.
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Notes
- 1.
This function can be constructed, for example, making use of suspension. Let \(f\in MS(M^n)\) and let \(\hat{f}^t\) be the flow on the manifold \(M^n\times \mathbb R\) induced by the vector field which consists of the unit vectors parallel to \(\mathbb R\) and directed to \(+\infty \). Define the diffeomorphism \(g:M^n\times \mathbb R\rightarrow M^n\times \mathbb R\) by \(g(x,\tau )=(f(x),\tau -1)\). Let \(G=\{g^k~,k\in \mathbb Z\}\) and \(W=(M^n\times \mathbb R)/G\). Let \(p_{_{W}}:M^n\times \mathbb R\rightarrow W\) denote the natural projection and let \(f^t\) denote the flow on the manifold W defined by \(f^t(x)=p_{_{W}}(\hat{f}^t(p^{-1}_{_{W}}(x)))\). The flow \(f^t\) is called the suspension over the diffeomorphism f. By construction the chain-recurrent set of the flow \(f^t\) consists of \(k_f\) periodic orbits \(\delta _i=p_{_{W}}(\mathscr {O}_i\times \mathbb R)~,i\in \{1,\dots , k_f\}\). Therefore, the suspension \(f^t\) is a Morse–Smale flow without fixed points. Then applying the results of [7] one constructs an energy function for the flow \(f^t\) whose restriction to \(M^n\) is the desired Lyapunov function for f.
- 2.
Notice that the conditions of Theorem 7.3 are not necessary. The paper [3] provides an example of a Morse–Smale diffeomorphism on the manifold \(\mathbb S^2\times \mathbb S^1\), which has a dynamically ordered energy function but the 1-dimensional attractor and the 1–dimensional repeller of it are not tightly embedded.
References
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Grines, V.Z., Medvedev, T.V., Pochinka, O.V. (2016). An Energy Function for Morse–Smale Diffeomorphisms on 3-Manifolds. In: Dynamical Systems on 2- and 3-Manifolds. Developments in Mathematics, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-44847-3_7
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