December  2020, 40(12): 6795-6813. doi: 10.3934/dcds.2020166

A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold

1. 

Department of Mathematical Sciences, Yeshiva University, New York, NY 10016, USA

2. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

3. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, 08028, Spain

* Corresponding author

Received  December 2019 Published  March 2020

Fund Project: M.G. was partially supported by NSF grant DMS-1814543.
R.L. was partially supported by NSF grant DMS-1800241.
T.S. was partially supported by the MINECO-FEDER Grant MTM2015-65715-P and PGC2018-098676-B-100, the Catalan Grant 2017SGR1049, and the ICREA Academia 2019 award.
Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester.

We describe a recent method to show instability in Hamiltonian systems. The main hypothesis of the method is that some explicit transversality conditions – which can be verified in concrete systems by finite calculations – are satisfied.

In particular, for several types of perturbations of integrable Hamiltonian systems, the hypothesis can be verified by just checking that some Melnikov-type integrals have non-degenerate zeros. This holds for Baire generic sets of perturbations in the $ C^r $-topology, for $ r \in [3, \infty) \cup \{\omega\} $. Our method does not require that the unperturbed Hamiltonian system is convex, or that the perturbation is polynomial, which are non-generic properties.

Provided that the transversality conditions are verified, one concludes the existence of orbits which change the action coordinate by a quantity independent of the size of the perturbation. In fact, one can obtain orbits that follow any path in action space, up to an error decreasing with the size of the perturbation.

Citation: Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
References:
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V. I. Arnold, Instability of dynamical systems with several degrees of freedom, in Vladimir I. Arnold - Collected Works, Collected Works, 1, Springer, Berlin, Heidelberg, 423–427. doi: 10.1007/978-3-642-01742-1_26.  Google Scholar

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P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998). doi: 10.1090/memo/0645.  Google Scholar

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P. W. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

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P. BernardV. Kaloshin and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders, Acta Math., 217 (2016), 1-79.  doi: 10.1007/s11511-016-0141-5.  Google Scholar

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M. Berti and P. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 395-450.  doi: 10.1016/S0294-1449(01)00084-1.  Google Scholar

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M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl. (9), 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.  Google Scholar

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S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063.  doi: 10.1088/0951-7715/19/9/003.  Google Scholar

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M. J. Capinski and M. Gidea, Arnold diffusion, quantitative estimates and stochastic behavior in the three-body problem, preprint, arXiv: 1812.03665. Google Scholar

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M. CapinskiM. Gidea and R. De la Llave, Arnold diffusion in the planar elliptic restricted three-body problem: Mechanism and numerical verification, Nonlinearity, 30 (2017), 329-360.  doi: 10.1088/1361-6544/30/1/329.  Google Scholar

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M. J. Capiński and P. Zgliczyński, Transition tori in the planar restricted elliptic three-body problem, Nonlinearity, 24 (2011), 1395-1432.  doi: 10.1088/0951-7715/24/5/002.  Google Scholar

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C.-Q. Cheng, Variational methods for the problem of Arnold diffusion, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,337–365. doi: 10.1007/978-1-4020-6964-2_14.  Google Scholar

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A. DelshamsR. de la Llave and T. M. Seara, Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion, Adv. Math., 294 (2016), 689-755.  doi: 10.1016/j.aim.2015.11.010.  Google Scholar

[22]

A. Delshams, M. Gidea, R. de la Llave and T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,285–336. doi: 10.1007/978-1-4020-6964-2_13.  Google Scholar

[23]

A. DelshamsM. Gidea and P. Roldan, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument, Phys. D, 334 (2016), 29-48.  doi: 10.1016/j.physd.2016.06.005.  Google Scholar

[24]

A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077.  doi: 10.1088/0951-7715/22/8/013.  Google Scholar

[25]

A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations, Regul. Chaotic Dyn., 22 (2017), 78-108.  doi: 10.1134/S1560354717010051.  Google Scholar

[26]

A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations with two independent harmonics, Discrete Contin. Dyn. Syst., 38 (2018), 6047-6072.  doi: 10.3934/dcds.2018261.  Google Scholar

[27]

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N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.  Google Scholar

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V. Gelfreich and D. Turaev, Unbounded energy growth in Hamiltonian systems with a slowly varying parameter, Comm. Math. Phys., 283 (2008), 769-794.  doi: 10.1007/s00220-008-0518-1.  Google Scholar

[31]

V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Comm. Math. Phys., 353 (2017), 507-547.  doi: 10.1007/s00220-017-2867-0.  Google Scholar

[32]

M. Gidea and R. de la Llave, Perturbations of geodesic flows by recurrent dynamics, J. Eur. Math. Soc. (JEMS), 19 (2017), 905-956.  doi: 10.4171/JEMS/683.  Google Scholar

[33]

M. Gidea and R. de la Llave, Global Melnikov theory in Hamiltonian systems with general time-dependent perturbations, J. Nonlinear Sci., 28 (2018), 1657-1707.  doi: 10.1007/s00332-018-9461-2.  Google Scholar

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M. Gidea, R. de la Llave and M. Musser, Global effect of non-conservative perturbations on homoclinic orbits, preprint, arXiv: 1909.02080. Google Scholar

[35]

M. Gidea, R. de la Llave and T. M-Seara, Accessibility and control in nearly integrable Hamiltonian systems, 2019. Google Scholar

[36]

M. GideaR. de la Llave and T. M-Seara, A general mechanism of diffusion in Hamiltonian systems: Qualitative results, Comm. Pure Appl. Math., 73 (2020), 150-209.  doi: 10.1002/cpa.21856.  Google Scholar

[37]

M. Gidea and J.-P. Marco, Diffusion along chains of normally hyperbolic cylinders, preprint, arXiv: 1708.08314. Google Scholar

[38]

A. Granados, Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators, Phys. D, 351/352 (2017), 14-29.  doi: 10.1016/j.physd.2017.04.003.  Google Scholar

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M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323.  Google Scholar

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M. GuzzoE. Lega and C. Froeschlé, A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems, Phys. D, 238 (2009), 1797-1807.  doi: 10.1016/j.physd.2009.06.009.  Google Scholar

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A. Luque and D. Peralta-Salas, Arnold diffusion of charged particles in ABC magnetic fields, J. Nonlinear Sci., 27 (2017), 721-774.  doi: 10.1007/s00332-016-9349-y.  Google Scholar

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J. N. Mather, Arnold diffusion. I. Announcement of results, J. Math. Sci. (N.Y.), 124 (2004), 5275-5289.  doi: 10.1023/B:JOTH.0000047353.78307.09.  Google Scholar

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J. N. Mather, Arnold diffusion by variational methods, in Essays in Mathematics and Its Applications, Springer, Heidelberg, 2012,271–285. doi: 10.1007/978-3-642-28821-0_11.  Google Scholar

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R. Moeckel, Generic drift on Cantor sets of annuli, in Celestial Mechanics, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002,163–171. doi: 10.1090/conm/292/04922.  Google Scholar

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D. Treschev, Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems, Nonlinearity, 15 (2002), 2033-2052.  doi: 10.1088/0951-7715/15/6/313.  Google Scholar

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show all references

References:
[1]

V. I. Arnold, Instability of dynamical systems with several degrees of freedom, in Vladimir I. Arnold - Collected Works, Collected Works, 1, Springer, Berlin, Heidelberg, 423–427. doi: 10.1007/978-3-642-01742-1_26.  Google Scholar

[2]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998). doi: 10.1090/memo/0645.  Google Scholar

[3]

P. W. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[4]

P. BernardV. Kaloshin and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders, Acta Math., 217 (2016), 1-79.  doi: 10.1007/s11511-016-0141-5.  Google Scholar

[5]

M. Berti and P. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 395-450.  doi: 10.1016/S0294-1449(01)00084-1.  Google Scholar

[6]

M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl. (9), 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.  Google Scholar

[7]

S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063.  doi: 10.1088/0951-7715/19/9/003.  Google Scholar

[8]

M. J. Capinski and M. Gidea, Arnold diffusion, quantitative estimates and stochastic behavior in the three-body problem, preprint, arXiv: 1812.03665. Google Scholar

[9]

M. CapinskiM. Gidea and R. De la Llave, Arnold diffusion in the planar elliptic restricted three-body problem: Mechanism and numerical verification, Nonlinearity, 30 (2017), 329-360.  doi: 10.1088/1361-6544/30/1/329.  Google Scholar

[10]

M. J. Capiński and P. Zgliczyński, Transition tori in the planar restricted elliptic three-body problem, Nonlinearity, 24 (2011), 1395-1432.  doi: 10.1088/0951-7715/24/5/002.  Google Scholar

[11]

M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order: I. The equation $y''- k (1-y^2) y'+ y = b\lambda k \cos(\lambda t+ \alpha)$, $k$ large, J. London Math. Soc., 20 (1945), 180-189.  doi: 10.1112/jlms/s1-20.3.180.  Google Scholar

[12]

Q. Chen and R. de la Llave, Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems, (2019). Google Scholar

[13]

C.-Q. Cheng, Variational methods for the problem of Arnold diffusion, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,337–365. doi: 10.1007/978-1-4020-6964-2_14.  Google Scholar

[14]

C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277.  doi: 10.4310/jdg/1246888485.  Google Scholar

[15]

L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144pp.  Google Scholar

[16]

B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52 (1979), 264-379.  doi: 10.1016/0370-1573(79)90023-1.  Google Scholar

[17]

W.-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011.  Google Scholar

[18]

A. DelshamsR. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T2, Comm. Math. Phys., 209 (2000), 353-392.  doi: 10.1007/PL00020961.  Google Scholar

[19]

A. DelshamsR. de la Llave and T. M Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math., 202 (2006), 64-188.  doi: 10.1016/j.aim.2005.03.005.  Google Scholar

[20]

A. DelshamsR. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153.  doi: 10.1016/j.aim.2007.08.014.  Google Scholar

[21]

A. DelshamsR. de la Llave and T. M. Seara, Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion, Adv. Math., 294 (2016), 689-755.  doi: 10.1016/j.aim.2015.11.010.  Google Scholar

[22]

A. Delshams, M. Gidea, R. de la Llave and T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in Hamiltonian Dynamical Systems and Applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, Dordrecht, 2008,285–336. doi: 10.1007/978-1-4020-6964-2_13.  Google Scholar

[23]

A. DelshamsM. Gidea and P. Roldan, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument, Phys. D, 334 (2016), 29-48.  doi: 10.1016/j.physd.2016.06.005.  Google Scholar

[24]

A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077.  doi: 10.1088/0951-7715/22/8/013.  Google Scholar

[25]

A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations, Regul. Chaotic Dyn., 22 (2017), 78-108.  doi: 10.1134/S1560354717010051.  Google Scholar

[26]

A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations with two independent harmonics, Discrete Contin. Dyn. Syst., 38 (2018), 6047-6072.  doi: 10.3934/dcds.2018261.  Google Scholar

[27]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

[28]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.  Google Scholar

[29]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346.  doi: 10.1017/S0143385700005575.  Google Scholar

[30]

V. Gelfreich and D. Turaev, Unbounded energy growth in Hamiltonian systems with a slowly varying parameter, Comm. Math. Phys., 283 (2008), 769-794.  doi: 10.1007/s00220-008-0518-1.  Google Scholar

[31]

V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Comm. Math. Phys., 353 (2017), 507-547.  doi: 10.1007/s00220-017-2867-0.  Google Scholar

[32]

M. Gidea and R. de la Llave, Perturbations of geodesic flows by recurrent dynamics, J. Eur. Math. Soc. (JEMS), 19 (2017), 905-956.  doi: 10.4171/JEMS/683.  Google Scholar

[33]

M. Gidea and R. de la Llave, Global Melnikov theory in Hamiltonian systems with general time-dependent perturbations, J. Nonlinear Sci., 28 (2018), 1657-1707.  doi: 10.1007/s00332-018-9461-2.  Google Scholar

[34]

M. Gidea, R. de la Llave and M. Musser, Global effect of non-conservative perturbations on homoclinic orbits, preprint, arXiv: 1909.02080. Google Scholar

[35]

M. Gidea, R. de la Llave and T. M-Seara, Accessibility and control in nearly integrable Hamiltonian systems, 2019. Google Scholar

[36]

M. GideaR. de la Llave and T. M-Seara, A general mechanism of diffusion in Hamiltonian systems: Qualitative results, Comm. Pure Appl. Math., 73 (2020), 150-209.  doi: 10.1002/cpa.21856.  Google Scholar

[37]

M. Gidea and J.-P. Marco, Diffusion along chains of normally hyperbolic cylinders, preprint, arXiv: 1708.08314. Google Scholar

[38]

A. Granados, Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators, Phys. D, 351/352 (2017), 14-29.  doi: 10.1016/j.physd.2017.04.003.  Google Scholar

[39]

M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian Geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323.  Google Scholar

[40]

M. GuzzoE. Lega and C. Froeschlé, A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems, Phys. D, 238 (2009), 1797-1807.  doi: 10.1016/j.physd.2009.06.009.  Google Scholar

[41]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.  Google Scholar

[42]

V. Kaloshin and K. Zhang, Arnold diffusion for smooth convex systems of two and a half degrees of freedom, Nonlinearity, 28 (2015), 2699-2720.  doi: 10.1088/0951-7715/28/8/2699.  Google Scholar

[43] A. Liapounoff, Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies, 17, Princeton University Press, Princeton, NJ; Oxford University Press, London, 1947.   Google Scholar
[44]

A. Luque and D. Peralta-Salas, Arnold diffusion of charged particles in ABC magnetic fields, J. Nonlinear Sci., 27 (2017), 721-774.  doi: 10.1007/s00332-016-9349-y.  Google Scholar

[45]

J. N. Mather, Arnold diffusion. I. Announcement of results, J. Math. Sci. (N.Y.), 124 (2004), 5275-5289.  doi: 10.1023/B:JOTH.0000047353.78307.09.  Google Scholar

[46]

J. N. Mather, Arnold diffusion by variational methods, in Essays in Mathematics and Its Applications, Springer, Heidelberg, 2012,271–285. doi: 10.1007/978-3-642-28821-0_11.  Google Scholar

[47]

R. Moeckel, Generic drift on Cantor sets of annuli, in Celestial Mechanics, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002,163–171. doi: 10.1090/conm/292/04922.  Google Scholar

[48]

Y. B. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.  Google Scholar

[49]

G. N. Piftankin, Diffusion speed in the Mather problem, Dokl. Akad. Nauk, 408 (2006), 736-737.  doi: 10.1088/0951-7715/19/11/007.  Google Scholar

[50]

H. Poincaré, Les Methodes Nouvelles de la Mecanique Celeste. Tome III, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987.  Google Scholar

[51]

P. K. Rashevsky, About connecting two points of a completely nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknechta, 2 (1938), 83-94.   Google Scholar

[52]

C. Robinson, Differentiable conjugacy near compact invariant manifolds, Bol. Soc. Brasil. Mat., 2 (1971), 33-44.  doi: 10.1007/BF02584805.  Google Scholar

[53] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, NJ, 1965.   Google Scholar
[54]

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