Legendrian persistance modules and dynamics (Q6601741)

From the authors' abstract: ``We relate the machinery of persistence modules to the Legendrian contact homology theory and to Poisson bracket invariants, and use it to show the existence of connecting trajectories of contact and symplectic Hamiltonian flows.''\N\NLet \((M,\omega)\) be a connected symletctic manifold. The symplectic gradient \(\mathrm{sgrad}\,H\) of a Hamiltonian \(H\) is defined via \(i_{\mathrm{sgrad}\, H}\omega=-dH\), and the Poisson bracket of two Hamiltonians \(F,G\) is given by \[\{F,G\}:=\omega(\mathrm{sgrad}\,G,\mathrm{sgrad}\,F)=dF(\mathrm{sgrad}\,G)=-dG(\mathrm{sgrad}\,F)=L_{\mathrm{sgrad}\,G}F=-L_{\mathrm{sgrad}\,F}G.\] Assume \(X_0,X_1,Y_0,Y_1\) are closed subsets of \(M\) such that \(X_0\cap X_1=Y_0\cap Y_1=\emptyset\). Such a collection of sets is called an admissible quadruple. For a pair \((F,G)\in C^\infty(M)\times C^\infty(M)\) and an admissible quadruple the authors prove four propositions (``Monotonicity'', ``Semi-continuity'', Prop. 2.6 and 2.7). Let \(\mathbb{I}:=(a,+\infty)\). A persistence module over \(\mathbb{I}\) is given by a pair \[(V=\{V_t\}_{t\in \mathbb{I}},\pi =\{\pi_{s.t}\in \mathbb{I },s\leq t\}),\] where all \(V_t,t\in \mathbb{I}\), are finite-dimensional \(\mathbb{Z}_2\)-vector space and \(\pi_{s,t}:V_s\rightarrow V_t\) are linear maps, so that:\N\begin{itemize}\N\item[(i)] (Persistence) \(\pi_{t,t}=Id, \pi_{s,r}=\pi_{t,r}\circ \pi{s,t}\), for \(s,t,r\in I,s\leq t\leq r\).\N\item[(ii)] (Discrete spectrum and semi-continuity). There exists a (finite or countable) discrete closed set of points\N\(\mathrm{spec}(V)=\{l_{\min}(V):=t_0<t_1<t_2<...<+\infty\}\subset \mathbb{I}\), called the spectrum of \(V\), so that\N\begin{itemize}\N\item[--] for any \(r\in \mathbb{I}\backslash \mathrm{spec}(V)\) , there exists a neughborhood \(U\) of \(r\in \mathbb{I}\) such that \(\pi_{s,t}\) is an isomorphism for all \(s,t\in U, s\leq t\);\N\item[--] for any \(r\in \mathrm{spec}(V)\), there exists \(\epsilon >0\) , such that \(\pi_{s,t}\) is an isomorphism of vector spaces for all \(s,t\in (r-\epsilon,r]\cap \mathbb{I}\).\N\end{itemize}\N\item[(iii)] (Semi-bounded support) For the smallest element \(l_{\min}:=t_0\) of \(\mathrm{spec}(V)\), one has \(l_{\min}(V)>a\) and \(V_t=0\) for all \(t\leq l_{\min}(V)\).\N\end{itemize}\NIn Section 3 the main result is the following.\N\NTheorem 3.4. For every persistence module \((V,\pi)\) over \(\mathbb{I}\) , there exists a unique (finite or countable) collection of intervals \(\mathbb{J}_j\subset \mathbb{I}\) - where intervals may not be distinct, but each interval appears in the collection only finitely many times - so that \((V,\pi)\) is isomorphic to \(\bigoplus_jQ(\mathbb{J}_j)\).\N\NThe collection \(\mathbb{J}_j\) of intervals is called the barcode of \(V\). The intervals are called the bars of \(V\).\N\NThe barcode of the trivial persistence module is empty. Set \(V_\infty=(\mathbb{Z}_2)^k\), where \(k\in \mathbb{N}\cup \{0,+\infty\}\) are the infinite bars in the barcode of \(V\). Other examples are also given.\N\NAnother method used in the article is a variant of a homology theory called Legendrian contact homology for which the authors cite papers of \textit{Y. Eliasberg}, \textit{M. Gromov}, \textit{Y. Chekanov}, and some papers by the authors themselves in preprint or in preparation. The statements of the theorems related to this theory (Theorems 4.9 and 4.10) are difficult to reproduce in this review. Section 5 entitled ``Applications to contact dynamics'' as well as the last two sections require their reading by specialists or well anchored in the field.\N\NFor the entire collection see [Zbl 1515.53004].