Abstract: In this paper, the problem of the existence of an energy function for discrete dynamical systems defined by structurally stable diffeomorphisms is studied. For the considered class of dynamical systems, such a function is defined as a continuous function that decreases along the orbits of the system outside the non-wandering set and remains constant on the non-wandering orbits. It is currently known that if the non-wandering set of a structurally stable surface diffeomorphism does not contain nontrivial saddle basic sets, then it possesses an energy function. However, if such a basic set exists and does not contain a pair of conjugated points, then the diffeomorphism does not admit such a function. The so-called serpentine geometric type of Markov partitioning is introduced, which generalizes the geometric type for the classical Smale horseshoe, and diffeomorphisms with nontrivial saddle basic sets admitting Markov partitions of this type are considered. Such basic sets contain pairs of conjugated points, accordingly, their study extends the class of diffeomorphisms for which the question of the existence of an energy function has been resolved. The main result of this paper is the proof that an orientation-preserving structurally stable 2-diffeomorphism with a basic set admitting a Markov partition of serpentine geometric type does not possess an energy function.
Keywords: energy function, structurally stable diffeomorphism, basic set, Markov partition, serpentine geometric type, pairs of conjugated points
Funding: The research was carried out within the framework of the Fundamental Research Program of the National Research University Higher School of Economics (Project № HSE-BR-2025-81).
For citation: Barinova, М. K. and Tirskaya, К. Yu. Nonexistence of Energy Function for Surface Diffeomorphisms with Nontrivial Basic Sets of Serpentine Geometric Type, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp. 25-43. DOI 10.46698/x0726-7762-7626-m
1. Conley, C. Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, no. 38, American Mathematical Society, 1978, 88 p.
2. Smale, S. On Gradient Dynamical Systems, Annals of Mathematics, 1961, vol. 74, no. 1, pp. 199-206. DOI: 10.2307/1970311.
3. Meyer, K. R. Energy Functions for Morse Smale Systems, American Journal of Mathematics, 1968, vol. 90, no. 4, pp. 1031-1040. DOI: 10.2307/2373287.
4. Franks, J. Nonsingular Smale Flows on S³, Topology, 1985, vol. 24, no. 3, pp. 265-282. DOI: 10.1016/0040-9383(85)90002-3.
5. Wilson, W. Smoothing Derivatives of Functions and Applications, Transactions of the American Mathematical Society, 1969, vol. 139, pp. 413-428. DOI: 10.2307/1995333.
6. Pixton, D. Wild Unstable Manifolds, Topology, 1977, vol. 16, no. 2, pp. 167-172. DOI: 10.1016/0040-9383(77)90014-3.
7. Grines, V. Z. and Kalai, Kh. Kh. Diffeomorphisms of Two-Dimensional Manifolds with Spatially Situated Basic Sets, Russian Mathematical Surveys, 1985, vol. 40, no. 1, pp. 221-222. DOI: 10.1070/RM1985v040n01ABEH003540.
8. Bonatti, C. and Langevin, R. Difféomorphismes de Smale des Surfaces, Astérisque, vol. 250, Paris, Société Mathématique de France, 1998, 235 p.
9. Béguin, F. Classification des Difféomorphismes de Smale des Surfaces: Types Géométriques Réalisables, Annales de l’Institut Fourier, 2002, vol. 52, no. 4, pp. 1135-1185. DOI: 10.5802/aif.1914.
10. Mitryakova, T. M., Pochinka, O. V. and Shishenkova, A. E. Energy Function for Diffeomorphisms on Surfaces with Finite Hyperbolic Chain Recurrent Set, Middle Volga Mathematical Society Journal, 2012, vol. 14, no. 1, pp. 98-106.
11. Grines, V. Z., Noskova, M. K. and Pochinka, O. V. The Construction of an Energy Function for Three-Dimensional Cascades with a Two-Dimensional Expanding Attractor, Transactions of the Moscow Mathematical Society, 2015, vol. 76, no. 2, pp. 237-249. DOI: 10.1090/mosc/249.
12. Barinova, M. On Existence of an Energy Function for Ω-Stable Surface Diffeomorphisms, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 14, pp. 3317-3323. DOI: 10.1134/S1995080222020020.
13. Barinova, M. K. and Tirskaya, K. Y. Nonexistence of Energy Function for Surface Diffeomorphisms with Horseshoe Basic Sets, Partial Differential Equations in Applied Mathematics, 2024, vol. 11. DOI: 10.1016/j.padiff.2024.100876.
14. Palis, J. J. and De Melo, W. Geometric Theory of Dynamical Systems: An Introduction, New York, Springer, 2012. DOI: 10.1007/978-1-4612-5703-5.
15. Shub, M. Stabilité Globale des Systèmes Dynamiques, Astérisque, vol. 56, Paris, Société Mathématique de France, 1978, 224 p.
16. Smale, S. The Ω-Stability Theorem, Global Analysis, Proceedings of Symposia in Pure Mathematics, 1970, vol. 14, pp. 289-297. DOI: 10.1090/pspum/014.
17. Franke, J. E. and Selgrade, J. F. Hyperbolicity and Chain Recurrence, Journal of Differential Equations, 1977, vol. 26, no. 1, pp. 27-36.
18. Robinson, C. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Boca Raton, FL, CRC-Press, 1999, 506 p.
19. Smale, S. Differentiable Dynamical Systems, Bulletin of the American Mathematical Society, 1967, vol. 73, no. 6, pp. 747-817. DOI: 10.1090/S0002-9904-1967-11798-1.
20. Grines, V. Z. and Pochinka, O. V. Construction of Energetic Functions for Ω-Stable Diffeomorphisms on 2- and 3-Manifolds, Contemporary Mathematics. Fundamental Directions, 2017, vol. 63, no. 2, pp. 191-222 (in Russian). DOI: 10.22363/2413-3639-2017-63-2-191-222.
21. Bowen, R. Topological Entropy and Axiom A, Global Analysis, American Mathematical Society, 1970, vol. 14, pp. 23-41.
22. Smale, S. A Structurally Stable Differentiable Homeomorphism with an Infinite Number of Periodic Points, Qualitative Methods in the Theory of Non-Linear Vibrations, Proc. Internat. Sympos. Non-linear Vibrations, 1961, vol. 2, pp. 365-366.
23. Grines, V. Z. The Topological Conjugacy of Diffeomorphisms of a Two-Dimensional Manifold on One-Dimensional Orientable Basic Sets. I, Transactions of the Moscow Mathematical Society, 1975, vol. 32, pp. 31-56.
24. Gonchenko, S. V., Gonchenko, A. S. and Malkin, M. I. On Classification of Classical and Half-Orientable Horseshoes in Terms of Boundary Points, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 549-566. DOI: 10.20537/nd1003006.
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