We prove that a certain C^{0}-robust condition of a Hamiltonian function H induces the existence of a point transported varepsilon out of the original point by the Hamiltonian diffeomorphism varphi_{H}. Related to our observation, we provide a problem on Hamiltonian pseudo-rotations

Kawasaki, Morimichi

Orita, Ryuma

English

Kyoto University Research Information Repository

TitleAPPLICATION OF FRAGMENTATION NORMS TOTRANSPORTED POINTS BY HAMILTONIAN ISOTOPIES(Geometry, Algebra and Combinatorics in Transformationgroup theory)Author(s)Kawasaki, Morimichi; Orita, RyumaCitation数理解析研究所講究録 = RIMS Kokyuroku (2018), 2098:57-59Issue Date2018-12URL http://hdl.handle.net/2433/251762RightType Departmental Bulletin PaperTextversionpublisherKyoto University57APPLICATION OF FRAGMENTATION NORMS TOTRANSPORTED POINTS BY HAMILTONIAN ISOTOPIESMORIMICHI KAWASAKI AND RYUMA ORITAABSTRACT. We prove that a certain C^{0}‐robust condition of a Hamiltonianfunction  H induces the existence of a point transported  \varepsilon out of the originalpoint by the Hamiltonian diffeomorphism  \varphi_{H} . Related to our observation, weprovide a problem on Hamiltonian pseudo‐rotations.1. MAIN RESULTLet  (M, \omega) be a symplectic manifold. For a Hamiltonian  H:S^{1}\cross Marrow \mathbb{R} withcompact support, we set  H_{t}=H(t, \cdot) for  t\in S^{1}=\mathbb{R}/\mathbb{Z} . The Hamiltonian vectorfield  X_{H_{t}} associated with  H_{t} is a time‐dependent vector field defined by the formula \omega(X_{H_{t}}, \cdot)=-dH_{t}.The Hamiltonian Isotopy  \{\varphi_{H}^{t}\}_{t\in \mathbb{R}} associated with  H is defined by \begin{array}{l}\varphi_{H}^{0}=id_{M},\frac{d}{dt}\varphi_{H}^{t}=X_{H_{t}}\circ\varphi_{H}^{t} for all t\in \mathbb{R},\end{array}and its time‐one map  \varphi_{H}=\varphi_{H}^{1} is referred to as the Hamiltonian diffeomorphism(with compact support) generated by  H . We denote by Ham(M) the group ofHamiltonian diffeomorphisms of  M with compact support.In his famous work [B], Banyaga proved the simplicity of Ham(M) when  M isclosed. The key ingredient was the proof of the fragmentation lemma for this group,which enables us to define fragmentation norms on Ham(M) as follows. Let  U be anopen subset of  M . The fragmentation lemma implies that for every  \phi\in Ham(M)there exists a positive integer  n such that  \phi can be represented as a product of n diffeomorphisms  \theta_{i}\in Ham(\psi_{i}(U)) , where  \psi_{i}\in Ham(M) and  1\leq i\leq n . For \phi\neq id_{M} , its fragmentation norm  \Vert\phi\Vert_{U} with respect to the open subset  U is definedto be the minimal number of factors in such a product. We set  \Vert\phi\Vert_{U}=0 when \phi=id_{M}.Definition 1.1. Let  U be an open subset of  M . The fragmentation norm  \Vert\cdot\Vert_{U} iscontrolled by the  C^{0} ‐topology if there exist an  C^{0}‐open neighborhood  \mathcal{U}\subset Ham(M)of the identity and a positive number  C>0 such that  \Vert\phi\Vert_{U}<C for any  \phi\in \mathcal{U}.Entov, Polterovich and Py proved the following interesting theorem.Theorem 1.2 ([EPP, Theorem 4]). Let  \Sigma be a compact Riemann surface (possiblywith boundary) equipped with an area form. Let   U\subset\Sigma be an open subset diffeo‐morphic to an open disc. Then the fragmentation norm  \Vert .  \Vert_{U} is controlled by the C^{0} ‐topology.Our main result is the following theorem which is an application of Theorem 1.2.Date: August 31, 2018.This work is supported by JSPS KAKENHI Grant Numbers  JP18J00765,  JP18J00335.58MORIMICHI KAWASAKI AND RYUMA ORITATheorem 1.3. Let  \Sigma_{g} be a closed Riemann surface of genus  g\geq 1 equipped withan area form and a Riemannian metric. Let  L_{1} and  L_{2} be non‐contractible simpleclosed curves in  \Sigma_{g} . Then there exist positive numbers  \varepsilon_{0}>0 and  C>0 such thatfor any Hamiltonian  H:S^{1}\cross\Sigma_{g}arrow \mathbb{R} satisfying  H|_{S^{{\imath}}\cross L_{1}}>C and  H|_{S^{{\imath}}\cross L_{2}}<0,there exists  x\in\Sigma_{g} such that  d(x, \varphi_{H}(x))>\varepsilon_{0}.Here  d is the distance induced by the Riemannian metric on  \Sigma_{g}.Remark 1.4. Since the Hamiltonians  kH,  k\in \mathbb{Z}_{>0} , also satisfy  kH|_{S^{1}\cross L_{1}}>C and kH|_{S^{1}\cross L_{2}}<0 , Theorem 1.3 yields that  \varphi_{H}^{k_{i}} never converges to the identity withrespect to the  C^{0}‐topology for any sequence  k_{i}arrow\infty.2. PROOF OF THEOREM 1.3To prove Theorem 1.3, we use the following theorem which is an application ofLagrangian spectral invariants [LZ,  K].Theorem 2.1 ([KO]). Let  \Sigma_{g} be a closed Riemann surface of genus  g\geq 1 equippedwith an area form and  U a contractible open subset of  \Sigma_{g} . Then, there existsa positive number  r satisfying the following condition. Let  L_{1} and  L_{2} be non‐contractible simple closed curves in  \Sigma_{g} . For any  C>0 and any Hamiltonian H:S^{1}\cross\Sigma_{g}arrow \mathbb{R} satisfying  H|_{S^{1}\cross L_{1}}>C and  H|_{S^{1}\cross L_{2}}<0, \Vert\varphi_{H}\Vert_{U}>r\cdot C.Proof of Theorem 1.3. Fix a Riemannian metric on  \Sigma_{g} . For a positive number  \varepsilon,we define a subset  \mathcal{U}_{\varepsilon} of Ham  (\Sigma_{g}) by \mathcal{U}_{\varepsilon}= {  \phi\in Ham(\Sigma_{g})|d(x, \phi(x))\leq\varepsilon for any  x\in\Sigma_{g} }.By Theorem 1.2, there exist positive numbers  C and  \varepsilon_{0} such that  \Vert\phi\Vert_{U}<C holdsfor any  \phi\in \mathcal{U}_{\varepsilon_{0}}.By Theorem 2.1, for any Hamiltonian  H:S^{1}\cross\Sigma_{g}arrow \mathbb{R} satisfying  H|_{S^{1}\cross L_{1}}>C/rand  H|_{S^{1}\cross L_{2}}<0, \Vert\varphi_{H}\Vert_{U}>r\cdot C/r=C.Then  \varphi_{H}\not\in \mathcal{U}_{\varepsilon_{0}} and hence, there exists a point  x in  \Sigma_{g} such that  d(x, \phi(x))>\varepsilon_{0}. \square 3. A PROBLEM ON HAMILTONIAN PSEUDO‐ROTATIONSLet  \mathbb{C}P^{n} be the  n‐dimensional complex projective space equipped with theFubini‐Study form  \omega_{FS}. A Hamiltonian pseudo‐rotation of  \mathbb{C}P^{n} is a Hamiltoniandiffeomorphism with exactly  n+1 fixed points. Observe that this is the minimalpossible number of fixed points of Hamiltonian diffeomorphisms of  \mathbb{C}P^{n} . Ginzburgand Gürel proved the following crucial theorem.Theorem 3.1 ([GG, Theorem 5.13]). Let  \varphi be a Hamiltonian pseudo‐rotation of \mathbb{C}P^{n} with exponentially Liouville mean index vector  \vec{A} (see [GG, Definition 5.11]for the definition). Then there exists a sequence   k_{i}arrow\infty such that \varphi^{k_{i}}arrow^{c^{0}} id.We consider  (\mathbb{C}P^{2}, \omega_{FS}) . The real projective space  \mathbb{R}P^{2} is naturally embedded in (\mathbb{C}P^{2}, \omega_{FS}) as a Lagrangian submanifold. There is another Lagrangian submanifold L_{W} called the Chekanov torus which is disjoint from  \mathbb{R}P^{2} . Then the authors provedthe following theorem.APPLICATION OF FRAGMENTATION NORMS59Theorem 3.2 ([KO]). Let  U be a displaceable open subset of  \mathbb{C}P^{2} . Then, thereexists a positive number  r satisfying the following condition. For any  C>0 andany Hamiltonian  H:S^{1}\cross \mathbb{C}P^{2}arrow \mathbb{R} satisfying  H|_{S^{{\imath}}\cross \mathbb{R}P^{2}}>C and  H|_{S^{{\imath}}\cross L_{W}}<0, \Vert\varphi_{H}\Vert_{U}>r\cdot C.Combining with Theorems 3.1 and 3.2, an argument similar to the proof ofTheorem 1.3 yields the following corollary.Corollary 3.3. Assume that the fragmentation norm  \Vert .  \Vert_{U} is controlled by the C^{0} ‐topology. Let  U be a displaceable open subset of  \mathbb{C}P^{2} and  H:S^{1}\cross \mathbb{C}P^{2}arrow \mathbb{R}a Hamiltonian satisfying  H|_{S^{1}\cross \mathbb{R}P^{2}}>C and  H|_{S^{1}\cross L_{W}}<0 . Then  \varphi_{H}\uparrow s not aHamiltonian pseudo‐rotation of  \mathbb{C}P^{2} with exponentially Liouville mean index vector \vec{\triangle}.Thus, we pose the following problem.Problem 3.4. Does there exist a positive number  C>0 such that for any Hamil‐tonian  H:S^{1}\cross \mathbb{C}P^{2}arrow \mathbb{R} satisfying  H|_{S^{1}x\mathbb{R}P^{2}}>C and  H|_{S^{1}\cross L_{W}}<0 , the Hamil‐tonian diffeomorphism  \varphi_{H} is not a Hamiltonian pseudo‐rotation of  \mathbb{C}P^{2} (i.e.,  \varphi_{H}has more than 3 fixed points) /?REFERENCES[B] A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme sym‐plectique, Comment. Math. Helv. 53 (1978) no. 2174‐227.[EPP] M. Entov, L. Polterovich and P. Py, On continuity of quasimorphisms for symplectic maps,With an appendix by Michael Khanevsky Progr. Math., 296, Perspectives in analysis, geom‐etry, and topology, 169‐197, Birkhäuser/Springer, New York (2012).[GG] V. Ginzburg and B. Gürel, Hamiltonian Pseudo‐rotations of Projective Spaces, to appearin Invent. Math., arXiv: 1712. 09766vl.[K] M. Kawasaki, Function theoretical applications of Lagrangian spectral invariants, in prepa‐ration.[KO] M. Kawasaki and R. Orita, Disjoint superheavy subsets and fragmentation norms,https://cgp.ibs.re.kr/archive/preprints/2018, Preprint (2018).[LZ] R. Leclercq and F. Zapolsky, Spectral invariants for monotone Lagrangians, J. Topol. Anal.,Online Ready (17 May 2017), https://doi.org/10.1142/S1793525318500267.(Morimichi Kawasaki) RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, KYOTO UNIVER‐SITY, KYOTO 606‐8502, JAPANE‐mail address: kawasaki@kurims.kyoto‐u.ac.jp(Ryuma Orita) DEPARTMENT OF MATHEMATICAL SCIENCES, TOKYO METROPOLITAN UNIVER‐SITY, TOKYO 192‐0397, JAPANE‐mail address: ryuma. orita@gmail.comURL: https://ryuma‐orita.github. io/

APPLICATION OF FRAGMENTATION NORMS TO TRANSPORTED POINTS BY HAMILTONIAN ISOTOPIES (Geometry, Algebra and Combinatorics in Transformation group theory)

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APPLICATION OF FRAGMENTATION NORMS TO TRANSPORTED POINTS BY HAMILTONIAN ISOTOPIES (Geometry, Algebra and Combinatorics in Transformation group theory)

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