October  2021, 8(4): 403-445. doi: 10.3934/jcd.2021016

Computing Reeb dynamics on four-dimensional convex polytopes

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Received  September 2020 Revised  July 2021 Published  October 2021 Early access  August 2021

Fund Project: JC was partially supported by an NSF Graduate Research Fellowship. MH was partially supported by NSF grant DMS-1708899, a Simons Fellowship, and a Humboldt Research Award

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $ 1 $.

Citation: Julian Chaidez, Michael Hutchings. Computing Reeb dynamics on four-dimensional convex polytopes. Journal of Computational Dynamics, 2021, 8 (4) : 403-445. doi: 10.3934/jcd.2021016
References:
[1]

A. AbbondandoloB. BramhamU. L. Hryniewicz and P. A. S. Salomão, Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., 211 (2018), 687-778.  doi: 10.1007/s00222-017-0755-z.  Google Scholar

[2]

A. AbbondandoloB. BramhamU. L. Hryniewicz and P. A. S. Salomão, Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere, Compositio Math., 154 (2018), 2643-2680.  doi: 10.1112/S0010437X18007558.  Google Scholar

[3]

A. Abbondandolo and J. Kang, Symplectic homology of convex domains and Clarke's duality, preprint, arXiv: 1907.07779. Google Scholar

[4]

S. Artstein-AvidanR. Karasev and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J., 163 (2014), 2003-2022.  doi: 10.1215/00127094-2794999.  Google Scholar

[5]

S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, IMRN, 2014, 165–193. doi: 10.1093/imrn/rns216.  Google Scholar

[6]

A. Balitskiy, Equiality cases in Viterbo's conjecture and isoperimeric billiard inequalities, Int. Math. Res. Not., 2020 (2020), 1957-1978.  doi: 10.1093/imrn/rny076.  Google Scholar

[7]

K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, In Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54 (2007), Cambridge University Press, 1–44. doi: 10.1017/CBO9780511755187.002.  Google Scholar

[8]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.  Google Scholar

[9]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II., Math Z., 203 (1990), 553-567.  doi: 10.1007/BF02570756.  Google Scholar

[10]

J. Gutt and M. Hutchings, Symplectic capacities from positive $S^1$-equivariant symplectic homology, Algebraic and Geometric Topology, 18 (2018), 3537-3600.  doi: 10.2140/agt.2018.18.3537.  Google Scholar

[11]

J. Gutt, M. Hutchings and V. G. B. Ramos, Examples around the strong Viterbo conjecture, preprint, arXiv: 2003.10854, to appear in Journal of Fixed Point Theory and Applications. Google Scholar

[12]

P. Haim-Kislev, On the symplectic size of convex polytopes, Geometric and Functional Analysis, 29 (2019), 440-463.  doi: 10.1007/s00039-019-00486-4.  Google Scholar

[13]

D. Hermann, Non-Equivalence of Symplectic Capacities for Open Sets with Restricted Contact Type Boundary., Prépublication d'Orsay numéro, 32 (1998). Google Scholar

[14]

H. HoferK. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. Math., 148 (1998), 197-289.  doi: 10.2307/120994.  Google Scholar

[15]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[16]

X. Hu and Y. Long, Closed characteristics on non-degenerate star-shaped hypersurfaces in $ {\mathbb R}^2n$, Science In China (Series A), 45 (2002), 1038-1052.  doi: 10.1007/BF02879987.  Google Scholar

[17]

M. Hutchings, Taubes's proof of the Weinstein conjecture in dimension three, Bull. AMS, 47 (2010), 73-125.  doi: 10.1090/S0273-0979-09-01282-8.  Google Scholar

[18]

M. Hutchings, Quantitative embedded contact homology, J. Diff. Geom., 88 (2011), 231-266.  doi: 10.4310/jdg/1320067647.  Google Scholar

[19]

U. Hryniewicz, private communication, 2017. Google Scholar

[20]

K. Irie, Symplectic homology of fiberwise convex sets and homology of loop spaces, arXiv: 1907.09749. Google Scholar

[21]

A. F. Künzle, Singular Hamiltonian systems and symplectic capacities, Singularities and Differential Equations, 171–187, Banach Center Publications 33, Polish Academy of Sciences, 1996.  Google Scholar

[22]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math, 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar

[23]

F. Schlenk, Embedding Problems in Symplectic Geometry, Walter de Gruyter, 2005. doi: 10.1515/9783110199697.  Google Scholar

[24]

K. Siegel, Higher symplectic capacities, preprint, arXiv: 1902.01490. Google Scholar

[25]

C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431.  doi: 10.1090/S0894-0347-00-00328-3.  Google Scholar

[26]

A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Diff. Geom, 9 (1974), 513-517.  doi: 10.4310/jdg/1214432547.  Google Scholar

show all references

References:
[1]

A. AbbondandoloB. BramhamU. L. Hryniewicz and P. A. S. Salomão, Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., 211 (2018), 687-778.  doi: 10.1007/s00222-017-0755-z.  Google Scholar

[2]

A. AbbondandoloB. BramhamU. L. Hryniewicz and P. A. S. Salomão, Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere, Compositio Math., 154 (2018), 2643-2680.  doi: 10.1112/S0010437X18007558.  Google Scholar

[3]

A. Abbondandolo and J. Kang, Symplectic homology of convex domains and Clarke's duality, preprint, arXiv: 1907.07779. Google Scholar

[4]

S. Artstein-AvidanR. Karasev and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J., 163 (2014), 2003-2022.  doi: 10.1215/00127094-2794999.  Google Scholar

[5]

S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, IMRN, 2014, 165–193. doi: 10.1093/imrn/rns216.  Google Scholar

[6]

A. Balitskiy, Equiality cases in Viterbo's conjecture and isoperimeric billiard inequalities, Int. Math. Res. Not., 2020 (2020), 1957-1978.  doi: 10.1093/imrn/rny076.  Google Scholar

[7]

K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, In Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54 (2007), Cambridge University Press, 1–44. doi: 10.1017/CBO9780511755187.002.  Google Scholar

[8]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.  Google Scholar

[9]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II., Math Z., 203 (1990), 553-567.  doi: 10.1007/BF02570756.  Google Scholar

[10]

J. Gutt and M. Hutchings, Symplectic capacities from positive $S^1$-equivariant symplectic homology, Algebraic and Geometric Topology, 18 (2018), 3537-3600.  doi: 10.2140/agt.2018.18.3537.  Google Scholar

[11]

J. Gutt, M. Hutchings and V. G. B. Ramos, Examples around the strong Viterbo conjecture, preprint, arXiv: 2003.10854, to appear in Journal of Fixed Point Theory and Applications. Google Scholar

[12]

P. Haim-Kislev, On the symplectic size of convex polytopes, Geometric and Functional Analysis, 29 (2019), 440-463.  doi: 10.1007/s00039-019-00486-4.  Google Scholar

[13]

D. Hermann, Non-Equivalence of Symplectic Capacities for Open Sets with Restricted Contact Type Boundary., Prépublication d'Orsay numéro, 32 (1998). Google Scholar

[14]

H. HoferK. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. Math., 148 (1998), 197-289.  doi: 10.2307/120994.  Google Scholar

[15]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[16]

X. Hu and Y. Long, Closed characteristics on non-degenerate star-shaped hypersurfaces in $ {\mathbb R}^2n$, Science In China (Series A), 45 (2002), 1038-1052.  doi: 10.1007/BF02879987.  Google Scholar

[17]

M. Hutchings, Taubes's proof of the Weinstein conjecture in dimension three, Bull. AMS, 47 (2010), 73-125.  doi: 10.1090/S0273-0979-09-01282-8.  Google Scholar

[18]

M. Hutchings, Quantitative embedded contact homology, J. Diff. Geom., 88 (2011), 231-266.  doi: 10.4310/jdg/1320067647.  Google Scholar

[19]

U. Hryniewicz, private communication, 2017. Google Scholar

[20]

K. Irie, Symplectic homology of fiberwise convex sets and homology of loop spaces, arXiv: 1907.09749. Google Scholar

[21]

A. F. Künzle, Singular Hamiltonian systems and symplectic capacities, Singularities and Differential Equations, 171–187, Banach Center Publications 33, Polish Academy of Sciences, 1996.  Google Scholar

[22]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math, 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar

[23]

F. Schlenk, Embedding Problems in Symplectic Geometry, Walter de Gruyter, 2005. doi: 10.1515/9783110199697.  Google Scholar

[24]

K. Siegel, Higher symplectic capacities, preprint, arXiv: 1902.01490. Google Scholar

[25]

C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431.  doi: 10.1090/S0894-0347-00-00328-3.  Google Scholar

[26]

A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Diff. Geom, 9 (1974), 513-517.  doi: 10.4310/jdg/1214432547.  Google Scholar

Figure 1.  We depict the tangent, normal and Reeb cones for two points $ p,q \in X $ in a polytope $ X \subset {\mathbb R}^2 $
Figure 2.  We depict sub-trajectories of the three types of orbits, in red. Each cube above represents a 3-face of a hypothetical 4-polytope
Figure 3.  An example of a flow graph with 4 nodes and 4 edges. The linear domains and flows are depicted above their corresponding nodes and edges
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