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Liouville type theorems for poly-harmonic Navier problems
Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator
1. | Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI 96822, United States |
2. | CERES-ERTI, École Normale Supérieure, 75005 Paris, France |
References:
[1] |
N. H. Abel, Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable,, in, 2 (1839), 246. Google Scholar |
[2] |
C. Aliprantis and K. Border, "Infinite Dimensional Analysis: A Hitchhiker's Guide,", Springer-Verlag, (2007). Google Scholar |
[3] |
R. Arens, Topologies for homeomorphism groups,, Amer J. Math., 68 (1946), 593.
|
[4] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advanced Series Nonlinear Dynamics, 16 (2000).
doi: 10.1142/9789812813633. |
[5] |
J. Banaś, A. Hajnosz and S. Wędrychowicz, On existence and asymptotic behavior of solutions of some functional equations,, Funkcialaj Ekvacioj, 25 (1982), 257.
|
[6] |
A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations and commutators of germs of contact diffeomorphisms,, Trans. Amer. Math. Society, 312 (1989), 755.
doi: 10.2307/2001010. |
[7] |
A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem,, J. Geom. Anal., 6 (1996), 613.
doi: 10.1007/BF02921624. |
[8] |
G. Belitskii and Yu. Lyubich, The Abel equation and total solvability of linear functional equations,, Studia Mathematica, 127 (1998), 81.
|
[9] |
G. Belitskii and Yu. Lyubich, The real-analytic solutions of the Abel functional equation,, Studia Mathematica, 134 (1999), 135.
|
[10] |
G. Belitskii and V. Tkachenko, Functional equations in real-analytic functions,, Studia Mathematica, 143 (2000), 153.
|
[11] |
P. S. Bourdon and J. H. Shapiro, Mean growth of Koenigs eigenfunctions,, J. Amer. Math. Soc., 10 (1997), 299.
doi: 10.1090/S0894-0347-97-00224-5. |
[12] |
J. Caugran and H. J. Schwartz, Spectra of compact composition operators,, Proc. Amer. Math. Soc., 51 (1975), 127.
|
[13] |
M. Chaperon, "Géométrie Différentielle et Singularités de Systèmes Dynamiques,", Astérisque, 138-139 (1986), 138.
|
[14] |
M. D. Chekroun, M. Ghil, J. Roux and F. Varadi, Averaging of time-periodic systems without a small parameter,, Disc. and Cont. Dyn. Syst. A, 14 (2006), 753.
doi: 10.3934/dcds.2006.14.753. |
[15] |
D. D. Clahane, Spectra of compact composition operators over bounded symmetric domains,, Integr. Equ. Oper. Theory, 51 (2005), 41.
doi: 10.1007/s00020-003-1250-z. |
[16] |
N. D. Cong, "Topological Dynamics of Random Dynamical Systems,", Oxford Mathematical Monographs, (1997).
|
[17] |
I. Cornfeld, S. Fomin and Ya. Sinaĭ, "Ergodic Theory,", Grundlehren der Mathematischen Wissenschaften, 245 (1982).
doi: 10.1007/978-1-4615-6927-5. |
[18] |
C. C. Cowen and B. D. MacCluer, "Composition Operators on Spaces of Analytic Functions,", Studies in Advanced Mathematics, (1995).
|
[19] |
R. de la Llave, J. Marko and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation,, Ann. of Math. (2), 123 (1986), 537.
doi: 10.2307/1971334. |
[20] |
M. A. Denjoy, Sur l'itération de fonctions analytiques,, C. R. Acad. Sci. Paris, 182 (1926), 255. Google Scholar |
[21] |
J. Dieudonné, "Éléments d'Analyse,", Tome 1, (1968). Google Scholar |
[22] |
J. Ding, The point spectrum of Perron-Frobenius and Koopman operators,, Proc. Amer. Math. Soc., 126 (1998), 1355.
doi: 10.1090/S0002-9939-98-04188-4. |
[23] |
G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems,, J. Dynam. Control Systems, 5 (1999), 173.
doi: 10.1023/A:1021726902801. |
[24] |
R. P. Gosselin, A maximal theorem for subadditive functions,, Acta Mathematica, 112 (1964), 163.
|
[25] |
M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976).
|
[26] |
M. C. Irwin, "Smooth Dynamical Systems,", Reprint of the 1980 original, 17 (1980).
doi: 10.1142/9789812810120. |
[27] |
G. Julia, Sur une classe d'équations fonctionnelles,, Annales Sci. de l'École Norm. Supérieure, 40 (1923), 97.
|
[28] |
R. R. Kallman, Uniqueness results for homeomorphism groups,, Trans. Amer. Math. Soc., 295 (1986), 389.
doi: 10.2307/2000162. |
[29] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Anatole Katok and Leonardo Mendoza, 54 (1995).
|
[30] |
J. L. Kelley, "General Topology,", Reprint of the 1955 edition [Van Nostrand, (1955).
|
[31] |
G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles,, Annales de l'École Normale Supérieure, 1 (1884), 3.
|
[32] |
B. Koopman and J. von Neumann, Dynamical systems of continuous spectra,, Proc. Nat. Acad. Sci. USA, 18 (1932), 255. Google Scholar |
[33] |
J. Kotus, M. Krych and Z. Nitecki, Global structural stability of flows on open surfaces,, Mem. Amer. Math. Soc., 37 (1982).
|
[34] |
M. Kuczma, "Functional Equations in a Single Variable,", Monografir Mat., 46 (1968).
|
[35] |
M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations,", Encyclopedia of Mathematics and its Applications, 32 (1990).
doi: 10.1017/CBO9781139086639. |
[36] |
A. Lasota and M. C. Mackey, "Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics,", Second edition, 97 (1994).
|
[37] |
A. Livshitz, Homology properties of $Y$-systems,, Math. Notes USSR Acad. Sci., 10 (1971), 758. Google Scholar |
[38] |
A. Livshitz, Cohomology of dynamical systems,, Math. USSR-Izv, 6 (1972), 1278. Google Scholar |
[39] |
R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon,, J. Phys. (Paris), 39 (1978), 69. Google Scholar |
[40] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behaviour,, Physica D, 197 (2004), 101.
doi: 10.1016/j.physd.2004.06.015. |
[41] |
M. Misiurewicz, Strange attractors for the Lozi mappings,, in, 357 (1980), 348.
|
[42] |
J. Palis, A global perspective for non-conservative dynamics,, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 22 (2005), 485.
doi: 10.1016/j.anihpc.2005.01.001. |
[43] |
R. A. Rosenbaum, Sub-additive functions,, Duke Math. J., 17 (1950), 227.
|
[44] |
J. Ren and X. Zhang, Topologies on homeomorphism spaces of certain metric spaces,, J. Math. Anal. Appl., 316 (2006), 32.
doi: 10.1016/j.jmaa.2005.05.019. |
[45] |
R. Roussarie and J. Roux, "Des Équations Différentielles aux Systèmes Dynamiques,", Tomes I et II, (2012). Google Scholar |
[46] |
H. H. Schaefer, "Topological Vector Spaces,", Second edition, 3 (1999). Google Scholar |
[47] |
E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen,, Math. Ann., 2 (1870), 317.
doi: 10.1007/BF01444024. |
[48] |
E. Seneta, Functional equations and the Galton-Watson process,, Advances in Applied Probability, 1 (1969), 1.
|
[49] |
J. H. Shapiro, W. Smith and D. A. Stegenga, Geometric models and compactness of composition operators,, J. Functional Analysis, 127 (1995), 21.
doi: 10.1006/jfan.1995.1002. |
[50] |
J. H. Shapiro, Composition operators and Schröder's functional equation,, in, 213 (1998), 213.
doi: 10.1090/conm/213/02861. |
[51] |
S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms,, in, (1963), 490.
|
[52] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
|
[53] |
J. Walorski, On the continuous smooth solutions of the Schröder equation in normed spaces,, Integr. Equ. Oper. Theory, 60 (2008), 597.
doi: 10.1007/s00020-007-1550-9. |
[54] |
J.-C. Yoccoz, Théorème de Siegel, nombre de Bruno et polynômes quadratiques,, Astérisque, 231 (1995), 3.
|
show all references
References:
[1] |
N. H. Abel, Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable,, in, 2 (1839), 246. Google Scholar |
[2] |
C. Aliprantis and K. Border, "Infinite Dimensional Analysis: A Hitchhiker's Guide,", Springer-Verlag, (2007). Google Scholar |
[3] |
R. Arens, Topologies for homeomorphism groups,, Amer J. Math., 68 (1946), 593.
|
[4] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advanced Series Nonlinear Dynamics, 16 (2000).
doi: 10.1142/9789812813633. |
[5] |
J. Banaś, A. Hajnosz and S. Wędrychowicz, On existence and asymptotic behavior of solutions of some functional equations,, Funkcialaj Ekvacioj, 25 (1982), 257.
|
[6] |
A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations and commutators of germs of contact diffeomorphisms,, Trans. Amer. Math. Society, 312 (1989), 755.
doi: 10.2307/2001010. |
[7] |
A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem,, J. Geom. Anal., 6 (1996), 613.
doi: 10.1007/BF02921624. |
[8] |
G. Belitskii and Yu. Lyubich, The Abel equation and total solvability of linear functional equations,, Studia Mathematica, 127 (1998), 81.
|
[9] |
G. Belitskii and Yu. Lyubich, The real-analytic solutions of the Abel functional equation,, Studia Mathematica, 134 (1999), 135.
|
[10] |
G. Belitskii and V. Tkachenko, Functional equations in real-analytic functions,, Studia Mathematica, 143 (2000), 153.
|
[11] |
P. S. Bourdon and J. H. Shapiro, Mean growth of Koenigs eigenfunctions,, J. Amer. Math. Soc., 10 (1997), 299.
doi: 10.1090/S0894-0347-97-00224-5. |
[12] |
J. Caugran and H. J. Schwartz, Spectra of compact composition operators,, Proc. Amer. Math. Soc., 51 (1975), 127.
|
[13] |
M. Chaperon, "Géométrie Différentielle et Singularités de Systèmes Dynamiques,", Astérisque, 138-139 (1986), 138.
|
[14] |
M. D. Chekroun, M. Ghil, J. Roux and F. Varadi, Averaging of time-periodic systems without a small parameter,, Disc. and Cont. Dyn. Syst. A, 14 (2006), 753.
doi: 10.3934/dcds.2006.14.753. |
[15] |
D. D. Clahane, Spectra of compact composition operators over bounded symmetric domains,, Integr. Equ. Oper. Theory, 51 (2005), 41.
doi: 10.1007/s00020-003-1250-z. |
[16] |
N. D. Cong, "Topological Dynamics of Random Dynamical Systems,", Oxford Mathematical Monographs, (1997).
|
[17] |
I. Cornfeld, S. Fomin and Ya. Sinaĭ, "Ergodic Theory,", Grundlehren der Mathematischen Wissenschaften, 245 (1982).
doi: 10.1007/978-1-4615-6927-5. |
[18] |
C. C. Cowen and B. D. MacCluer, "Composition Operators on Spaces of Analytic Functions,", Studies in Advanced Mathematics, (1995).
|
[19] |
R. de la Llave, J. Marko and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation,, Ann. of Math. (2), 123 (1986), 537.
doi: 10.2307/1971334. |
[20] |
M. A. Denjoy, Sur l'itération de fonctions analytiques,, C. R. Acad. Sci. Paris, 182 (1926), 255. Google Scholar |
[21] |
J. Dieudonné, "Éléments d'Analyse,", Tome 1, (1968). Google Scholar |
[22] |
J. Ding, The point spectrum of Perron-Frobenius and Koopman operators,, Proc. Amer. Math. Soc., 126 (1998), 1355.
doi: 10.1090/S0002-9939-98-04188-4. |
[23] |
G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems,, J. Dynam. Control Systems, 5 (1999), 173.
doi: 10.1023/A:1021726902801. |
[24] |
R. P. Gosselin, A maximal theorem for subadditive functions,, Acta Mathematica, 112 (1964), 163.
|
[25] |
M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976).
|
[26] |
M. C. Irwin, "Smooth Dynamical Systems,", Reprint of the 1980 original, 17 (1980).
doi: 10.1142/9789812810120. |
[27] |
G. Julia, Sur une classe d'équations fonctionnelles,, Annales Sci. de l'École Norm. Supérieure, 40 (1923), 97.
|
[28] |
R. R. Kallman, Uniqueness results for homeomorphism groups,, Trans. Amer. Math. Soc., 295 (1986), 389.
doi: 10.2307/2000162. |
[29] |
A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by Anatole Katok and Leonardo Mendoza, 54 (1995).
|
[30] |
J. L. Kelley, "General Topology,", Reprint of the 1955 edition [Van Nostrand, (1955).
|
[31] |
G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles,, Annales de l'École Normale Supérieure, 1 (1884), 3.
|
[32] |
B. Koopman and J. von Neumann, Dynamical systems of continuous spectra,, Proc. Nat. Acad. Sci. USA, 18 (1932), 255. Google Scholar |
[33] |
J. Kotus, M. Krych and Z. Nitecki, Global structural stability of flows on open surfaces,, Mem. Amer. Math. Soc., 37 (1982).
|
[34] |
M. Kuczma, "Functional Equations in a Single Variable,", Monografir Mat., 46 (1968).
|
[35] |
M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations,", Encyclopedia of Mathematics and its Applications, 32 (1990).
doi: 10.1017/CBO9781139086639. |
[36] |
A. Lasota and M. C. Mackey, "Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics,", Second edition, 97 (1994).
|
[37] |
A. Livshitz, Homology properties of $Y$-systems,, Math. Notes USSR Acad. Sci., 10 (1971), 758. Google Scholar |
[38] |
A. Livshitz, Cohomology of dynamical systems,, Math. USSR-Izv, 6 (1972), 1278. Google Scholar |
[39] |
R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon,, J. Phys. (Paris), 39 (1978), 69. Google Scholar |
[40] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behaviour,, Physica D, 197 (2004), 101.
doi: 10.1016/j.physd.2004.06.015. |
[41] |
M. Misiurewicz, Strange attractors for the Lozi mappings,, in, 357 (1980), 348.
|
[42] |
J. Palis, A global perspective for non-conservative dynamics,, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 22 (2005), 485.
doi: 10.1016/j.anihpc.2005.01.001. |
[43] |
R. A. Rosenbaum, Sub-additive functions,, Duke Math. J., 17 (1950), 227.
|
[44] |
J. Ren and X. Zhang, Topologies on homeomorphism spaces of certain metric spaces,, J. Math. Anal. Appl., 316 (2006), 32.
doi: 10.1016/j.jmaa.2005.05.019. |
[45] |
R. Roussarie and J. Roux, "Des Équations Différentielles aux Systèmes Dynamiques,", Tomes I et II, (2012). Google Scholar |
[46] |
H. H. Schaefer, "Topological Vector Spaces,", Second edition, 3 (1999). Google Scholar |
[47] |
E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen,, Math. Ann., 2 (1870), 317.
doi: 10.1007/BF01444024. |
[48] |
E. Seneta, Functional equations and the Galton-Watson process,, Advances in Applied Probability, 1 (1969), 1.
|
[49] |
J. H. Shapiro, W. Smith and D. A. Stegenga, Geometric models and compactness of composition operators,, J. Functional Analysis, 127 (1995), 21.
doi: 10.1006/jfan.1995.1002. |
[50] |
J. H. Shapiro, Composition operators and Schröder's functional equation,, in, 213 (1998), 213.
doi: 10.1090/conm/213/02861. |
[51] |
S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms,, in, (1963), 490.
|
[52] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
|
[53] |
J. Walorski, On the continuous smooth solutions of the Schröder equation in normed spaces,, Integr. Equ. Oper. Theory, 60 (2008), 597.
doi: 10.1007/s00020-007-1550-9. |
[54] |
J.-C. Yoccoz, Théorème de Siegel, nombre de Bruno et polynômes quadratiques,, Astérisque, 231 (1995), 3.
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