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On the growth rate of leaf-wise intersections. (English) Zbl 1266.53076

Let \(T^*M\) be the cotangent bundle of a closed connected orientable manifold \(M\). Then, a closed connected hypersurface \(\Sigma\) is said to a fiberwise starshaped hypersurface, if \(\lambda_\Sigma= \lambda|_\Sigma\), where \(\Sigma\) is the Liouville 1-form on \(T^*M\), is a positive contact form on \(\Sigma\). \(p\in\Sigma\) is said to be a leaf-wise intersection point for a compactly supported Hamiltonian diffeomorphisms \(\varphi\), if \[ \varphi(\phi^\Sigma_\eta(p))= p\quad\text{for some }\eta\in\mathbb R. \] Here, \(\phi:\Sigma\to\Sigma\) denotes the flow of \(R_\Sigma\), the Reeb vector field associated to \(\lambda|_\Sigma\). In [J. Moser, Acta Math. 141, 17–34 (1978; Zbl 0382.53035)], it was asked whether a given Hamiltonian diffeomorphism always has a leaf-wise intersection point in a given fibrewise starshaped surface, and if so, whether one can obtain a lower bound on the number of such leaf-wise intersections [P. Albers and U. Frauenfelder, J. Topol. Anal. 2, No. 1, 77–98 (2010; Zbl 1196.53050)].
In this paper, it is shown that if the homology of the free loop space \(\Lambda M\) of \(M\), \(\dim M\geq 2\), is sufficiently complicated, then not only generic Hamiltonian diffeomorphisms have infinitely many leaf-wise intersection points in any non-degenerate fiberwise starshaped hypersurface, but also the number of such leaf-wise intersection points “grows” exponentially with time (Theorem A, Corollary B and C). Examples of such manifolds are \((S^2\times S^2)\sharp (S^2\times S^2)\), \(\mathbb T^4\sharp\mathbb C P^2\), or any surface of genus greater than one (Prop.1.7).
These results extend results of P. Albers and U. Frauenfelder [Expo. Math. 30, No. 2, 168–181 (2012; Zbl 1272.53079)]. In this paper, the existence of infinitely many leaf-wise intersection points if the homology of \(\Lambda M\) is infinite-dimensional is proved by using Rabinowitz-Floer homology. But the authors say that the exponential growth of leaf-wise intersection point is the first result which establishes the existence of “more” than just infinitely many intersection points.
Let \(F\in C^\infty(T^* M,\mathbb R)\) be an autonomous Hamiltonian, \(\chi\in C^\infty(S^1,[0, 1])\) and let \(F^\chi= \chi(t)F(x)\). Then for a fixed \(f\in C^\infty(\mathbb R, \mathbb R)\), the Rabinowitz action functional \(A_{F^\chi,f}\) is defined by \[ A_{F^\chi,f}(x,\eta)= \int x^*\lambda- f(\eta) \int^1_0 f^\chi(t, x)\,dt. \] Rabinowitz-Floer homology is defined by using the Rabinowitz action functional [K. Cieliebak, U. Frauenfelder and A. Oancea, Ann. Sci. Éc. Norm. Supér. (4) 43, No. 6, 957–1015 (2010; Zbl 1213.53105)]. But it is not known whether Rabinowitz homology behaves well with respect to monotone homotopies [K. Cieliebak and U. Frauenfelder, Pac. J. Math. 239, No. 2, 251–316 (2009; Zbl 1221.53112)]. To overcome this difficulty, the authors introduce a perturbed Rabinowitz action functional \(A^H_{f^\chi,f}= A_{{\mathfrak f}}\) associated to the quadruple \({\mathfrak f}= (F,f,\chi,H)\), \(H\in C^\infty(S^1\times T^* M,\mathbb R)\), by \[ A_{{\mathfrak f}}(x,\eta)= A_{F^\chi, f}(x,\eta)- \int^1_0 H(t,x)\,dt, \] (§3.1). Then, a new variant of Rabinowitz-Floer homology \(HF^{(a,b)} (A_{{\mathfrak f}},\alpha)\), \(\alpha\in [S^1,M]\), \(a< b\leq\infty\), is constructed, by using \(A_{{\mathfrak f}}\) (§3.2 and §3.6. Coefficients of homologies are always \(\mathbb Z_2\)). After introducing the notion of admissible quadruple, it is shown that this new variant of Rabinowitz-Floer homology groups behaves well with respect to monotone homotopies [P. Biran, I. Polterovich and D. Salamon, Duke Math. J. 119, No. 1, 65–118 (2003; Zbl 1034.53089), Theorem 4.2]. As a consequence, the number \(n_{\Sigma,\alpha}(\phi,(a, b))\) of positive leaf-wise intersection points of \(\phi\) in \(\Sigma\) that belong to \(\alpha\) and have time shift \(T\in(a, b)\) is shown to be at least \[ \text{rank}\{i: HF^{(a-\| H\|_-,b-\| H\|_-)}(A_{{\mathfrak h}},\alpha)\to HF^{(a+\| H\|_+, b+\| H\|_+)}(A_{{\mathfrak h}},\alpha)\}. \] Here, \({\mathfrak h}= (F,f,1,0)\) and \[ \| H\|_+= \int^1_0 \max_{(q,p)\in T^* M}dt,\quad\| H\|_-= \int^1_0 \min_{(q,p)\in T^* M} H(t,q,p)\,dt. \] On the other hand, adopting Morse-Smale theory, for a certain class of \({\mathfrak f}\), isomorphisms \[ HF^{(c,\infty)}(A_{{\mathfrak f}}, \alpha)\cong\begin{cases} H(\Lambda_\alpha M),\quad &\alpha\neq 0,\\ H(\Lambda_0 M,M),\quad &\alpha= 0.\end{cases} \] are proved. Here, \(c\) is a certain number and \(\Lambda_\alpha(M)\) means the space of loops of \(M\) whose homotopy class is \(\alpha\) [A. Abbondandolo and M. Schwarz, J. Topol. Anal. 1, No. 4, 307–405 (2009; Zbl 1190.53082), Theorem 5.13]. Equality of ranks of some maps between (new variant of) Rabinowitz-Floer homologies and homologies of some kind of loop spaces are also proved.
By these facts, it is shown, assuming \(\dim M\geq 2\), that if \(\phi\) is a Hamiltonian diffeomorphism having periodic leaf-wise intersection points and the corresponding Rabinowitz action functional is Morse, and \(T\) is sufficiently large, then \(n_{(\Sigma,\alpha)}(\phi, T)\) is at least \[ \text{rank}\{\iota: H((\Lambda^{c(T-\|\phi\|)}_\alpha(M, g))\to H(\Lambda_\alpha M, \Lambda^{4\mu(\phi)}_\alpha(M, g)\}, \] if \(\alpha\neq 0\) and \[ \text{rank}\{\iota: H(\Lambda^{c(T-\|\phi\|)}_0(M, g)\to H(\Lambda_0 M, \Lambda^{4\mu(\phi)}_0 (M, g)\}, \] if \(\alpha= 0\). Here, \(g\) is a Riemannian metric on \(M\) such that \(S^*_g M\) is non-degenerate and contained in the interior of the compact region bounded by \(\Sigma\), \(c>0\) is a certain number, and \[ \Lambda^a_\alpha M= \Biggl\{q\in \Lambda_\alpha M;\,\int^1_0 |\dot q|^2\,dt\leq a^2\Biggr\}. \] As corollaries, exponentially growth of \(n_{\Sigma,\alpha}(\phi,T)\) with \(T\) if \(M\) is \((\Lambda,\alpha)\)-energy hyperbolic or \(\overline\pi_1(M)\) has exponential growth are shown. Here, \(M\) is said to be \((\Lambda,\alpha)\)-energy hyperbolic; if \[ \liminf_{a\to\infty}\, {\log\text{rank}\{\iota: H(\Lambda^a_\alpha(M, g))\to H(\Lambda_\alpha M)\}\over a}> 0, \] and \(\overline\pi_1(M)\cong [S^1, M]\), the fundamental group of \(M\) modulo conjugacy classes, is said to have exponential growth if \[ \lim_{k\to\infty}\, {\log\gamma_{{\mathcal S}}(k)\over k}> 0, \] with \[ \gamma_{{\mathcal S}}(k)= \sharp\{\alpha\in \overline\pi_1(M): \exists s_1,\dots, s_k\in{\mathcal S}\cup{\mathcal S}^{-1},\,\alpha= \overline{s_1\cdots s_k}\}, \] where \({\mathcal S}\subseteq \pi_1(M)\) is a finite set of generators.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
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