Solving the cohomological equation for locally Hamiltonian flows. I: Local obstructions. (English) Zbl 07842513
The authors study the cohomological equation \(Xu = f\) for smooth locally Hamiltonian flows on compact surfaces. Cohomological equations are an important area of study in dynamical systems since they are related to smooth conjugacy problems via Kolmogorov-Arnold-Moser techniques.
The main novelty of the proposed approach is that it is used to study the regularity of the solution \(u \) when the flow has saddle loops, which has not been systematically studied before. The existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs) is proved. The main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni’s distributions (see [G. Forni, Ann. Math. (2) 146, No. 2, 295–344 (1997; Zbl 0893.58037); Ann. Math. (2) 155, No. 1, 1–103 (2002; Zbl 1034.37003)]), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of \(f\) around the saddles. The main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a. locally Hamiltonian flows (with or without saddle loops) contained in [the authors, “Solving the cohomological equation for locally Hamiltonian flows, part II – global obstructions”, Preprint, arXiv:2306.02340]. The authors develop some new tools of handling functions whose higher derivatives have polynomial singularities over IETs.
The main novelty of the proposed approach is that it is used to study the regularity of the solution \(u \) when the flow has saddle loops, which has not been systematically studied before. The existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs) is proved. The main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni’s distributions (see [G. Forni, Ann. Math. (2) 146, No. 2, 295–344 (1997; Zbl 0893.58037); Ann. Math. (2) 155, No. 1, 1–103 (2002; Zbl 1034.37003)]), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of \(f\) around the saddles. The main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a. locally Hamiltonian flows (with or without saddle loops) contained in [the authors, “Solving the cohomological equation for locally Hamiltonian flows, part II – global obstructions”, Preprint, arXiv:2306.02340]. The authors develop some new tools of handling functions whose higher derivatives have polynomial singularities over IETs.
Reviewer: Zdzisław Dzedzej (Gdańsk)
MSC:
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 37J12 | Fixed points and periodic points of finite-dimensional Hamiltonian and Lagrangian systems |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37E35 | Flows on surfaces |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
References:
| [1] | Arnold, V. I., Topological and ergodic properties of closed 1-forms with incommensurable periods, Funkc. Anal. Prilozh., 25, 1-12, 1991, translation in Funct. Anal. Appl. 25 (1991), 81-90 · Zbl 0732.58001 |
| [2] | Avila, A.; Fayad, B.; Kocsard, A., On manifolds supporting distributionally uniquely ergodic diffeomorphisms, J. Differ. Geom., 99, 191-213, 2015 · Zbl 1316.37015 |
| [3] | Avila, A.; Kocsard, A., Cohomological equations and invariant distributions for minimal circle diffeomorphisms, Duke Math. J., 158, 501-536, 2011 · Zbl 1225.37052 |
| [4] | Bufetov, A., Limit theorems for translation flow, Ann. Math. (2), 179, 431-499, 2014 · Zbl 1290.37023 |
| [5] | Chaika, J.; Wright, A., A smooth mixing flow on a surface with nondegenerate fixed points, J. Am. Math. Soc., 32, 81-117, 2019 · Zbl 1454.37025 |
| [6] | Fayad, B.; Kanigowski, A.; Zelada, R., A non-mixing Arnold flow on a surface, preprint |
| [7] | Flaminio, L.; Forni, G., Invariant distributions and time averages for horocycle flows, Duke Math. J., 119, 465-526, 2003 · Zbl 1044.37017 |
| [8] | Flaminio, L.; Forni, G., On the cohomological equation for nilflows, J. Mod. Dyn., 1, 37-60, 2007 · Zbl 1114.37004 |
| [9] | Forni, G., Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. Math. (2), 146, 295-344, 1997 · Zbl 0893.58037 |
| [10] | Forni, G., Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math. (2), 155, 1-103, 2002 · Zbl 1034.37003 |
| [11] | Forni, G., Sobolev regularity of solutions of the cohomological equation, Ergod. Theory Dyn. Syst., 41, 685-789, 2021 · Zbl 1515.37034 |
| [12] | Forni, G., Twisted cohomological equations for translation flows, Ergod. Theory Dyn. Syst., 42, 881-916, 2022 · Zbl 1503.37056 |
| [13] | Forni, G.; Marmi, S.; Matheus, C., Cohomological equation and local conjugacy class of Diophantine interval exchange maps, Proc. Am. Math. Soc., 2024, in press |
| [14] | Frączek, K.; Kim, M., New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows, J. Reine Angew. Math., 807, 81-149, 2024 · Zbl 1539.37067 |
| [15] | Frączek, K.; Kim, M., Solving the cohomological equation for locally Hamiltonian flows, part II - global obstructions, preprint |
| [16] | Frączek, K.; Ulcigrai, C., Ergodic properties of infinite extensions of area-preserving flows, Math. Ann., 354, 1289-1367, 2012 · Zbl 1280.37004 |
| [17] | Frączek, K.; Ulcigrai, C., On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions, Comment. Math. Helv., 99, 231-354, 2024 · Zbl 1542.37003 |
| [18] | Giulietti, P.; Liverani, C., Parabolic dynamics and anisotropic Banach spaces, J. Eur. Math. Soc., 21, 2793-2858, 2019 · Zbl 1425.37005 |
| [19] | Katok, A., Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, vol. 30, 2003, American Mathematical Society: American Mathematical Society Providence, RI, iv+121 pp · Zbl 1030.37001 |
| [20] | Kochergin, A. V., Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, Mat. Sb. (N. S.), 96, 138, 471-502, 1975 · Zbl 0321.28012 |
| [21] | Marmi, S.; Moussa, P.; Yoccoz, J.-C., The cohomological equation for Roth-type interval exchange maps, J. Am. Math. Soc., 18, 823-872, 2005 · Zbl 1112.37002 |
| [22] | Marmi, S.; Moussa, P.; Yoccoz, J.-C., Linearization of generalized interval exchange maps, Ann. Math. (2), 176, 1583-1646, 2012 · Zbl 1277.37067 |
| [23] | Marmi, S.; Yoccoz, J.-C., Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps, Commun. Math. Phys., 344, 117-139, 2016 · Zbl 1360.37101 |
| [24] | Novikov, S. P., The Hamiltonian formalism and a multivalued analogue of Morse theory, Usp. Mat. Nauk, 37, 3-49, 1982, translated in Russian Math. Surveys 37 (1982), 1-56 · Zbl 0571.58011 |
| [25] | Ravotti, D., Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces, Ann. Henri Poincaré, 18, 3815-3861, 2017 · Zbl 1417.37189 |
| [26] | Roussarie, R., Modèles locaux de champs et de formes, Astérisqûe, vol. 30, 1975, Société Mathématique de France: Société Mathématique de France Paris, 181 pp · Zbl 0327.57017 |
| [27] | Tanis, J., The cohomological equation and invariant distributions for horocycle maps, Ergod. Theory Dyn. Syst., 34, 299-340, 2014 · Zbl 1302.37024 |
| [28] | Ulcigrai, C., Mixing of asymmetric logarithmic suspension flows over interval exchange transformations, Ergod. Theory Dyn. Syst., 27, 991-1035, 2007 · Zbl 1135.37004 |
| [29] | Ulcigrai, C., Weak mixing for logarithmic flows over interval exchange transformations, J. Mod. Dyn., 3, 35-49, 2009 · Zbl 1183.37081 |
| [30] | Ulcigrai, C., Absence of mixing in area-preserving flows on surfaces, Ann. Math. (2), 173, 1743-1778, 2011 · Zbl 1251.37003 |
| [31] | Ulcigrai, C., Dynamics and ‘arithmetics’ of higher genus surface flows, (ICM Proceedings, 2022) |
| [32] | Wang, Z. J., Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geom. Funct. Anal., 25, 1956-2020, 2015 · Zbl 1331.22012 |
| [33] | Zorich, A., How do the leaves of a closed 1-form wind around a surface?, (Pseudoperiodic Topology. Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2, 197, Adv. Math. Sci., vol. 46, 1999, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 135-178 · Zbl 0976.37012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
