December  2019, 39(12): 7057-7080. doi: 10.3934/dcds.2019295

On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

1. 

School of Mathematical Sciences, Beijing Normal University, No. 19, Xinjiekouwai St., Haidian District, Beijing 100875, China

2. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry St. Atlanta GA 30332, USA

To Luis Caffarelli in his 70th birthday, with admiration.

Received  October 2018 Revised  June 2019 Published  September 2019

Fund Project: X. S. supported by the National Natural Science Foundation of China (Grant No. 11871242).
R. L. has been partially supported by NSF grant DMS-1800241. Progress was made while R.L. was visiting the JLU-GT institute for theoretical Science and Beijing Normal University. The final version was written while R.L. was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester supported by DMS-1440140

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [30,Section 5.11] to illustrate the need of certain definitions in the calculus of variations.

The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics.

In the language of geodesics, the Almgren-Federer example constructs metrics in $ \mathbb{S}^1\times \mathbb{S}^2 $, with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class.

In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in $ \mathbb{T}^3 $ for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers.

For dynamics, the example also illustrates different definitions of "integrable" and clarifies the relation between minimization and hyperbolicity and its interaction with topology.

Citation: Xifeng Su, Rafael de la Llave. On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295
References:
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V. Bangert, Mather sets for twist maps and geodesics on tori, Dynam. Report. Ser. Dynam. Systems Appl., Wiley, Chichester, 1 (1988), 1-56. Google Scholar

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V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal., 9 (1999), 413-427. doi: 10.1007/s000390050093. Google Scholar

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V. Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems, 10 (1990), 263-286. doi: 10.1017/S014338570000554X. Google Scholar

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P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533–1568, http://aif.cedram.org/item?id=AIF_2002__52_5_1533_0. doi: 10.5802/aif.1924. Google Scholar

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R. de la Llave and E. Valdinoci, Ground states and critical points for Aubry-Mather theory in statistical mechanics, J. Nonlinear Sci., 20 (2010), 153-218. doi: 10.1007/s00332-009-9055-0. Google Scholar

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A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652. doi: 10.1016/S0764-4442(97)84777-5. Google Scholar

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A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, 2008.Google Scholar

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H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. Google Scholar

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H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2011, Reprint of the 1994 edition. doi: 10.1007/978-3-0348-0104-1. Google Scholar

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H. KochR. de la Llave and C. Radin, Aubry-Mather theory for functions on lattices, Discrete Contin. Dynam. Systems, 3 (1997), 135-151. doi: 10.3934/dcds.1997.3.135. Google Scholar

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P. Le Calvez, Les ensembles d'Aubry-Mather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 51-54. Google Scholar

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M. Levi, Shadowing property of geodesics in Hedlund's metric, Ergodic Theory Dynam. Systems, 17 (1997), 187-203. doi: 10.1017/S0143385797060999. Google Scholar

[37]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638. doi: 10.1088/0951-7715/5/3/001. Google Scholar

[38]

R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310. doi: 10.1088/0951-7715/9/2/002. Google Scholar

[39]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 141-153. doi: 10.1007/BF01233389. Google Scholar

[40]

R. Mañé, Global Variational Methods in Conservative Dynamics, Instituto de Matemática pura e aplicada, 1990.Google Scholar

[41]

J. N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4. Google Scholar

[42]

J. N. Mather, Minimal measures, Comment. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683. Google Scholar

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J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383. Google Scholar

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J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, in Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991), vol. 1589 of Lecture Notes in Math., Springer, Berlin, 1994, 92–186. doi: 10.1007/BFb0074076. Google Scholar

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D. Offin, Hyperbolic minimizing geodesics, Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1. Google Scholar

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G. P. Paternain, Geodesic Flows, vol. 180 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1. Google Scholar

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I. C. Percival, Variational principles for the invariant toroids of classical dynamics, J. Phys. A, 7 (1974), 794-802. doi: 10.1088/0305-4470/7/7/005. Google Scholar

[53]

I. C. Percival, A variational principle for invariant tori of fixed frequency, J. Phys. A, 12 (1979), L57–L60. doi: 10.1088/0305-4470/12/3/001. Google Scholar

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P. H. Rabinowitz and E. W. Stredulinsky, Extensions of Moser-Bangert Theory, vol. 81 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser/Springer, New York, 2011, Locally minimal solutions. doi: 10.1007/978-0-8176-8117-3. Google Scholar

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A. Sorrentino, Action-minimizing Methods in Hamiltonian Dynamics, vol. 50 of Mathematical Notes, Princeton University Press, Princeton, NJ, 2015, An introduction to Aubry-Mather theory. doi: 10.1515/9781400866618. Google Scholar

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X. Su and R. de la Llave, KAM theory for quasi-periodic equilibria in one-dimensional quasi-periodic media, SIAM J. Math. Anal., 44 (2012), 3901-3927. doi: 10.1137/12087160X. Google Scholar

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show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978, Second edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman. Google Scholar

[2]

M.-C. Arnaud, Lyapunov exponents for conservative twisting dynamics: A survey, in Ergodic theory, De Gruyter, Berlin, 2016,108–133. Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, [1989?], Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition. Google Scholar

[4]

V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Translated from the French by A. Avez, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Google Scholar

[5]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6. Google Scholar

[6]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynam. Report. Ser. Dynam. Systems Appl., Wiley, Chichester, 1 (1988), 1-56. Google Scholar

[7]

V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal., 9 (1999), 413-427. doi: 10.1007/s000390050093. Google Scholar

[8]

V. Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems, 10 (1990), 263-286. doi: 10.1017/S014338570000554X. Google Scholar

[9]

P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble), 52 (2002), 1533–1568, http://aif.cedram.org/item?id=AIF_2002__52_5_1533_0. doi: 10.5802/aif.1924. Google Scholar

[10]

U. BessiL. Chierchia and E. Valdinoci, Upper bounds on Arnold diffusion times via Mather theory, J. Math. Pures Appl. (9), 80 (2001), 105-129. doi: 10.1016/S0021-7824(00)01188-0. Google Scholar

[11]

M. L. Blank, Metric properties of minimal solutions of discrete periodical variational problems, Nonlinearity, 2 (1989), 1–22, http://stacks.iop.org/0951-7715/2/1. doi: 10.1088/0951-7715/2/1/001. Google Scholar

[12]

S. Bolotin, Homoclinic trajectories of invariant sets of Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 359-389. doi: 10.1007/s000300050020. Google Scholar

[13]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Chapman & Hall/CRC, Boca Raton, FL, 2004, Geometry, topology, classification, Translated from the 1999 Russian original. doi: 10.1201/9780203643426. Google Scholar

[14]

K. S. Brown, Cohomology of Groups, vol. 87 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994, Corrected reprint of the 1982 original. Google Scholar

[15]

L. A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math., 54 (2001), 1403-1441. doi: 10.1002/cpa.10008. Google Scholar

[16]

L. A. Caffarelli and R. de la Llave, Interfaces of ground states in Ising models with periodic coefficients, J. Stat. Phys., 118 (2005), 687-719. doi: 10.1007/s10955-004-8825-1. Google Scholar

[17]

A. Candel and R. de la Llave, On the Aubry-Mather theory in statistical mechanics, Comm. Math. Phys., 192 (1998), 649-669. doi: 10.1007/s002200050313. Google Scholar

[18]

C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965. Google Scholar

[19]

M. J. D. Carneiro, On minimizing measures of the action of autonomous Lagrangians, Nonlinearity, 8 (1995), 1077-1085. doi: 10.1088/0951-7715/8/6/011. Google Scholar

[20]

G. ContrerasA. Figalli and L. Rifford, Generic hyperbolicity of Aubry sets on surfaces, Invent. Math., 200 (2015), 201-261. doi: 10.1007/s00222-014-0533-0. Google Scholar

[21]

G. ContrerasJ. Delgado and R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits. Ⅱ, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 155-196. doi: 10.1007/BF01233390. Google Scholar

[22]

G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22° Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. Google Scholar

[23]

G. Contreras and G. P. Paternain, Connecting orbits between static classes for generic Lagrangian systems, Topology, 41 (2002), 645-666. doi: 10.1016/S0040-9383(00)00042-2. Google Scholar

[24]

X. CuiC.-Q. Cheng and W. Cheng, Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems, J. Differential Equations, 214 (2005), 176-188. doi: 10.1016/j.jde.2004.08.008. Google Scholar

[25]

R. de la Llave and N. P. Petrov, Theory of circle maps and the problem of one-dimensional optical resonator with a periodically moving wall, Phys. Rev. E (3), 59 (1999), 6637-6651. doi: 10.1103/PhysRevE.59.6637. Google Scholar

[26]

R. de la Llave and E. Valdinoci, Ground states and critical points for Aubry-Mather theory in statistical mechanics, J. Nonlinear Sci., 20 (2010), 153-218. doi: 10.1007/s00332-009-9055-0. Google Scholar

[27]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 649-652. doi: 10.1016/S0764-4442(97)84777-5. Google Scholar

[28]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, 2008.Google Scholar

[29]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. Google Scholar

[30]

H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J., 24 (1974/75), 351-407. doi: 10.1512/iumj.1975.24.24031. Google Scholar

[31]

G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. (2), 33 (1932), 719-739. doi: 10.2307/1968215. Google Scholar

[32]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2011, Reprint of the 1994 edition. doi: 10.1007/978-3-0348-0104-1. Google Scholar

[33]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995, With a supplementary chapter by Katok and Leonardo Mendoza. doi: 10.1017/CBO9780511809187. Google Scholar

[34]

H. KochR. de la Llave and C. Radin, Aubry-Mather theory for functions on lattices, Discrete Contin. Dynam. Systems, 3 (1997), 135-151. doi: 10.3934/dcds.1997.3.135. Google Scholar

[35]

P. Le Calvez, Les ensembles d'Aubry-Mather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 51-54. Google Scholar

[36]

M. Levi, Shadowing property of geodesics in Hedlund's metric, Ergodic Theory Dynam. Systems, 17 (1997), 187-203. doi: 10.1017/S0143385797060999. Google Scholar

[37]

R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638. doi: 10.1088/0951-7715/5/3/001. Google Scholar

[38]

R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310. doi: 10.1088/0951-7715/9/2/002. Google Scholar

[39]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 141-153. doi: 10.1007/BF01233389. Google Scholar

[40]

R. Mañé, Global Variational Methods in Conservative Dynamics, Instituto de Matemática pura e aplicada, 1990.Google Scholar

[41]

J. N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4. Google Scholar

[42]

J. N. Mather, Minimal measures, Comment. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683. Google Scholar

[43]

J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383. Google Scholar

[44]

J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, in Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991), vol. 1589 of Lecture Notes in Math., Springer, Berlin, 1994, 92–186. doi: 10.1007/BFb0074076. Google Scholar

[45]

M. Mazzucchelli, Critical Point Theory for Lagrangian Systems, vol. 293 of Progress in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0163-8. Google Scholar

[46]

B. M. McCoy, Advanced Statistical Mechanics, vol. 146 of International Series of Monographs on Physics, Oxford University Press, Oxford, 2010. Google Scholar

[47]

H. M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc., 26 (1924), 25–60, http://dx.doi.org/10.2307/1989225. doi: 10.1090/S0002-9947-1924-1501263-9. Google Scholar

[48]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229–272, http://www.numdam.org/item?id=AIHPC_1986__3_3_229_0. doi: 10.1016/S0294-1449(16)30387-0. Google Scholar

[49]

J. Moser, Selected Chapters in the Calculus of Variations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003, Lecture notes by Oliver Knill. doi: 10.1007/978-3-0348-8057-2. Google Scholar

[50]

D. Offin, Hyperbolic minimizing geodesics, Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1. Google Scholar

[51]

G. P. Paternain, Geodesic Flows, vol. 180 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1. Google Scholar

[52]

I. C. Percival, Variational principles for the invariant toroids of classical dynamics, J. Phys. A, 7 (1974), 794-802. doi: 10.1088/0305-4470/7/7/005. Google Scholar

[53]

I. C. Percival, A variational principle for invariant tori of fixed frequency, J. Phys. A, 12 (1979), L57–L60. doi: 10.1088/0305-4470/12/3/001. Google Scholar

[54]

P. H. Rabinowitz and E. W. Stredulinsky, Extensions of Moser-Bangert Theory, vol. 81 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser/Springer, New York, 2011, Locally minimal solutions. doi: 10.1007/978-0-8176-8117-3. Google Scholar

[55]

R. C. Robinson, An Introduction to Dynamical Systems–-Continuous and Discrete, vol. 19 of Pure and Applied Undergraduate Texts, 2nd edition, American Mathematical Society, Providence, RI, 2012. Google Scholar

[56]

A. Sorrentino, Action-minimizing Methods in Hamiltonian Dynamics, vol. 50 of Mathematical Notes, Princeton University Press, Princeton, NJ, 2015, An introduction to Aubry-Mather theory. doi: 10.1515/9781400866618. Google Scholar

[57]

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Figure 1.  Heuristic picture of the Almgren-Federer metric for a fixed x
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