Geodesic
Criterion for the Existence of an Energy Function for a Regular Homeomorphism of the 3-Sphere
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 45-61.

Voir la notice de l'article provenant de la source Math-Net.Ru

C. Conley's fundamental theorem of the theory of dynamical systems states that every dynamical system, even a nonsmooth one (i.e., a continuous flow or a discrete dynamical system generated by a homeomorphism), admits a continuous Lyapunov function. A Lyapunov function is strictly decreasing along the trajectories of the dynamical system outside the chain recurrent set and is constant on the chain component. A Lyapunov function whose set of critical points coincides with the chain recurrent set of the dynamical system is called an energy function; it has the closest relationship with the dynamics. However, not every dynamical system has an energy function. In particular, according to D. Pixton, even a structurally stable diffeomorphism with nonwandering set consisting of four fixed points may not have a smooth energy function. Our main result in this paper is a criterion for the existence of a continuous Morse energy function for regular homeomorphisms of the $3$-sphere, according to which the existence of such a function is equivalent to the asymptotic triviality of one-dimensional saddle manifolds. The criterion generalizes the results of V. Z. Grines, F. Laudenbach, and O. V. Pochinka for Morse–Smale $3$-diffeomorphisms in the case when the ambient manifold is the three-dimensional sphere. In particular, our criterion implies that Pixton's examples do not admit even a continuous energy function. 
@article{TM_2023_321_a2,
     author = {M. K. Barinova and V. Z. Grines and O. V. Pochinka},
     title = {Criterion for the {Existence} of an {Energy} {Function} for a {Regular} {Homeomorphism} of the {3-Sphere}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {45--61},
     publisher = {mathdoc},
     volume = {321},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2023_321_a2/}
}
TY  - JOUR
AU  - M. K. Barinova
AU  - V. Z. Grines
AU  - O. V. Pochinka
TI  - Criterion for the Existence of an Energy Function for a Regular Homeomorphism of the 3-Sphere
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2023
SP  - 45
EP  - 61
VL  - 321
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2023_321_a2/
LA  - ru
ID  - TM_2023_321_a2
ER  - 
%0 Journal Article
%A M. K. Barinova
%A V. Z. Grines
%A O. V. Pochinka
%T Criterion for the Existence of an Energy Function for a Regular Homeomorphism of the 3-Sphere
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2023
%P 45-61
%V 321
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2023_321_a2/
%G ru
%F TM_2023_321_a2
M. K. Barinova; V. Z. Grines; O. V. Pochinka. Criterion for the Existence of an Energy Function for a Regular Homeomorphism of the 3-Sphere. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 45-61. http://geodesic.mathdoc.fr/item/TM_2023_321_a2/

[1] Bonatti C., Grines V.Z., “Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dyn. Control Syst., 6:4 (2000), 579–602 | DOI | MR | Zbl

[2] Brown M., “A proof of the generalized Schoenflies theorem”, Bull. Amer. Math. Soc., 66:2 (1960), 74–76 | DOI | MR | Zbl

[3] Conley C., Isolated invariant sets and the Morse index, Reg. Conf. Ser. Math., 38, Amer. Math. Soc., Providence, RI, 1978 | MR | Zbl

[4] Fox R.H., Artin E., “Some wild cells and spheres in three-dimensional space”, Ann. Math. Ser. 2, 99 (1948), 979–990 | DOI | MR

[5] V. Z. Grines, F. Laudenbach, and O. V. Pochinka, “Dynamically ordered energy function for Morse–Smale diffeomorphisms on 3-manifolds”, Proc. Steklov Inst. Math., 278 (2012), 27–40 | DOI | MR | Zbl

[6] Grines V.Z., Medvedev T.V., Pochinka O.V., Dynamical systems on 2- and 3-manifolds, Dev. Math., 46, Springer, Cham, 2016 | MR | Zbl

[7] Harrold O.G., \textup {Jr.}, Griffith H.C., Posey E.E., “A characterization of tame curves in three-space”, Trans. Amer. Math. Soc., 79:1 (1955), 12–34 | DOI | MR | Zbl

[8] Hurewicz W., Wallman H., Dimension theory, Princeton Math. Ser., 4, Princeton Univ. Press, Princeton, NJ, 2015 | MR

[9] Kirby R.C., Siebenmann L.C., Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Stud., 88, Princeton Univ. Press, Princeton, NJ, 2016 | MR

[10] Kosniowski C., A first course in algebraic topology, Cambridge Univ. Press, Cambridge, 1980 | MR | Zbl

[11] Mazur B., “A note on some contractible 4-manifolds”, Ann. Math. Ser. 2, 73 (1961), 221–228 | DOI | MR | Zbl

[12] Medvedev T.V., Pochinka O.V., “The wild Fox–Artin arc in invariant sets of dynamical systems”, Dyn. Syst., 33:4 (2018), 660–666 | DOI | MR | Zbl

[13] Medvedev T.V., Pochinka O.V., Zinina S.K., “On existence of Morse energy function for topological flows”, Adv. Math., 378 (2021), 107518 | DOI | MR | Zbl

[14] Meyer K.R., “Energy functions for Morse–Smale systems”, Amer. J. Math., 90:4 (1968), 1031–1040 | DOI | MR | Zbl

[15] Mitryakova T.M., Pochinka O.V., Shishenkova A.E., “Energeticheskaya funktsiya dlya diffeomorfizmov poverkhnostei s konechnym giperbolicheskim tsepno rekurrentnym mnozhestvom”, Zhurn. Srednevolzhsk. mat. o-va, 14:1 (2012), 98–106 | Zbl

[16] Moise E.E., “Affine structures in 3-manifolds. V: The triangulation theorem and Hauptvermutung”, Ann. Math. Ser. 2, 56 (1952), 96–114 | DOI | MR | Zbl

[17] Moise E.E., Geometric topology in dimensions 2 and 3, Grad. Texts Math., 47, Springer, New York, 1977 | DOI | MR | Zbl

[18] Morse M., “Topologically non-degenerate functions on a compact $n$-manifold $M$”, J. anal. math., 7 (1959), 189–208 | DOI | MR | Zbl

[19] Pixton D., “Wild unstable manifolds”, Topology, 16 (1977), 167–172 | DOI | MR | Zbl

[20] Quinn F., “Topological transversality holds in all dimensions”, Bull Amer. Math. Soc., 18:2 (1988), 145–148 | DOI | MR | Zbl

[21] Smale S., “On gradient dynamical systems”, Ann. Math. Ser. 2, 74 (1961), 199–206 | DOI | MR | Zbl

[22] Szpilrajn E., “Sur l'extension de l'ordre partiel”, Fundam. math., 16 (1930), 386–389 | DOI