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To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (7) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your question, though there might exist. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (37) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your question, though there might exist. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (7) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your questions. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (7) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your question, though there might exist. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (7) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\lbrace \lbrace f\leq k+\frac{1}{2}\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your questions. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory,

C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978.

Conley and Zehnder, in Sect. 3 in their paper Morse type index theory for flows ..., Comm. Pure Appl. Math. (7) (1984), have shown how to extend the Morse inequalities to this more general case. This extension is based on a certain increasing filtration of the space $X$ compatible in a certain way with the flow $\Phi_t$. Such a filtration produces a spectral sequence in homology and the Morse inequalities relate the Betti numbers of the $E_1$ page of this spectral sequence to the Betti numbers of $X$. $\newcommand{\bZ}{\mathbb{Z}}$

In the very special case when $X$ is a compact smooth manifold, and the flow in question is the (negative) gradient flow of a self-indexing Morse function satisfying the Smale transversality conditions several nice things happen.

1. The above Conley-Zehnder filtration can be identified with the filtration by the sublevel sets $\bigl\lbrace f\leq k+\frac{1}{2}\bigr\rbrace$, $k\in\bZ_{\geq 0}$. (I recall that the critical values of $f$ are nonnegative integers.)
2. The $E_1$ page of the canonical spectral sequence of this filtration is the Floer complex; see sec. 2.5 of these lectures.

I have not seen applications of the Conley index applied directly to Hamiltonian flows as suggested in your questions. At a first sight this seems difficult but, as Arnold liked to say, you never know how hard a problem is until you try to solve it.