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

Xifeng Su; Rafael de la Llave

doi:10.3934/dcds.2019295

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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

Xifeng Su ^1, and Rafael de la Llave ^2,

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.

Keywords: Global calculus of variation, geodesics, c-minimizers, flat chains, periodic orbits.

Mathematics Subject Classification: Primary: 37J50; Secondary: 49Q15, 34C25, 53C22, 37J35.

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 and Continuous Dynamical Systems, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295

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.
[2]	M.-C. Arnaud, Lyapunov exponents for conservative twisting dynamics: A survey, in Ergodic theory, De Gruyter, Berlin, 2016,108–133.
[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.
[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.
[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.
[6]	V. Bangert, Mather sets for twist maps and geodesics on tori, Dynam. Report. Ser. Dynam. Systems Appl., Wiley, Chichester, 1 (1988), 1-56.
[7]	V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal., 9 (1999), 413-427. doi: 10.1007/s000390050093.
[8]	V. Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems, 10 (1990), 263-286. doi: 10.1017/S014338570000554X.
[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.
[10]	U. Bessi, L. 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.
[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.
[12]	S. Bolotin, Homoclinic trajectories of invariant sets of Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 359-389. doi: 10.1007/s000300050020.
[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.
[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.
[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.
[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.
[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.
[18]	C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965.
[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.
[20]	G. Contreras, A. Figalli and L. Rifford, Generic hyperbolicity of Aubry sets on surfaces, Invent. Math., 200 (2015), 201-261. doi: 10.1007/s00222-014-0533-0.
[21]	G. Contreras, J. 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.
[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.
[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.
[24]	X. Cui, C.-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.
[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.
[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.
[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.
[28]	A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, 2008.
[29]	H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.
[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.
[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.
[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.
[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.
[34]	H. Koch, R. 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.
[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.
[36]	M. Levi, Shadowing property of geodesics in Hedlund's metric, Ergodic Theory Dynam. Systems, 17 (1997), 187-203. doi: 10.1017/S0143385797060999.
[37]	R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638. doi: 10.1088/0951-7715/5/3/001.
[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.
[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.
[40]	R. Mañé, Global Variational Methods in Conservative Dynamics, Instituto de Matemática pura e aplicada, 1990.
[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.
[42]	J. N. Mather, Minimal measures, Comment. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683.
[43]	J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[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.
[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.
[46]	B. M. McCoy, Advanced Statistical Mechanics, vol. 146 of International Series of Monographs on Physics, Oxford University Press, Oxford, 2010.
[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.
[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.
[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.
[50]	D. Offin, Hyperbolic minimizing geodesics, Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1.
[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.
[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.
[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.
[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.
[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.
[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.
[57]	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.
[58]	X. Su and R. de la Llave, Percival Lagrangian approach to the Aubry-Mather theory, Expo. Math., 30 (2012), 182-208. doi: 10.1016/j.exmath.2012.01.003.

show all references

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

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.
[2]	M.-C. Arnaud, Lyapunov exponents for conservative twisting dynamics: A survey, in Ergodic theory, De Gruyter, Berlin, 2016,108–133.
[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.
[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.
[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.
[6]	V. Bangert, Mather sets for twist maps and geodesics on tori, Dynam. Report. Ser. Dynam. Systems Appl., Wiley, Chichester, 1 (1988), 1-56.
[7]	V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal., 9 (1999), 413-427. doi: 10.1007/s000390050093.
[8]	V. Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems, 10 (1990), 263-286. doi: 10.1017/S014338570000554X.
[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.
[10]	U. Bessi, L. 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.
[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.
[12]	S. Bolotin, Homoclinic trajectories of invariant sets of Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 359-389. doi: 10.1007/s000300050020.
[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.
[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.
[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.
[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.
[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.
[18]	C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965.
[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.
[20]	G. Contreras, A. Figalli and L. Rifford, Generic hyperbolicity of Aubry sets on surfaces, Invent. Math., 200 (2015), 201-261. doi: 10.1007/s00222-014-0533-0.
[21]	G. Contreras, J. 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.
[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.
[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.
[24]	X. Cui, C.-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.
[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.
[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.
[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.
[28]	A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, 2008.
[29]	H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.
[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.
[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.
[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.
[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.
[34]	H. Koch, R. 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.
[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.
[36]	M. Levi, Shadowing property of geodesics in Hedlund's metric, Ergodic Theory Dynam. Systems, 17 (1997), 187-203. doi: 10.1017/S0143385797060999.
[37]	R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638. doi: 10.1088/0951-7715/5/3/001.
[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.
[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.
[40]	R. Mañé, Global Variational Methods in Conservative Dynamics, Instituto de Matemática pura e aplicada, 1990.
[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.
[42]	J. N. Mather, Minimal measures, Comment. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683.
[43]	J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.
[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.
[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.
[46]	B. M. McCoy, Advanced Statistical Mechanics, vol. 146 of International Series of Monographs on Physics, Oxford University Press, Oxford, 2010.
[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.
[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.
[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.
[50]	D. Offin, Hyperbolic minimizing geodesics, Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1.
[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.
[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.
[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.
[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.
[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.
[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.
[57]	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.
[58]	X. Su and R. de la Llave, Percival Lagrangian approach to the Aubry-Mather theory, Expo. Math., 30 (2012), 182-208. doi: 10.1016/j.exmath.2012.01.003.

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

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2021 Impact Factor: 1.588

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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