Regularity of the composition operator in spaces of hölder functions

Discrete and Continuous Dynamical Systems

Volumn 5, Issue 1, 1999, Pages 157-184

  De La Llave, R ^a   Obaya, R ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF VALLADOLID   (Spain)

Author keywords

Composition operator; Differentiability properties; H lder spaces

Indexed keywords

EID: 0033477150     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.1999.5.157     Document Type: Article

References (41)

1. R. ABRAHAM, J. ROBBIN, Transversal Mappings and flows, Benjamin, New York, 1967.
2. J. APPEL, P. ZABREJKO, Nonlinear superposition operator, Cambridge University Press, 1990.
3. V. I. ARNOL'D, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1996), 319-361.
4. V. I. AVERBUKH, O. G. SMOLYANOV, The various definitions of the derivative in linear topological spaces, Russian Math. Surveys 23, 4 (1968), 67-113.
5. A. BANYAGA, Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique, Comm. Math. Helv. 53 (1978), 174-277.
6. A.M. BLOCH, H. FLASCHKA, T.S. RATIU, The Toda PDE and the geometry of the diffeomorphism group of the annulus, Fields Inst. Comm. 7 (1992), 57-92.
7. R. DEVILLE, G GODEFROY, V. ZIZLER Smoothness and renormings in Banach spaces Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific, 1993.
8. D.G. EBIN, J. MARSDEN, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163.
9. J. EELS, A setting for global analysis, Bull. A.M.S. 72 (1966), 751-807.
10. M. FRECHET, La notion de différentielle dans l'analyse général, C. R. Acad. Sci. Paris 180 (1925), 806-809.
11. D.B.A. EPSTEIN, The simplicity of certain groups of homeomorphisms, Compos. Math. 22 (1972), 165-173.
12. J. HADAMARD, Sur le module maximum d'une fonction et de ses derives, Bull. Soc. Math. France 42 (1898), 68-72.
13. R. HAMILTON, The inverse function theorem of Nash and Moser, Bull. A.M.S. 7 (1982), 65-222.
14. M. P. HEBLE, Aproximation of differentiate functions on a Hilbert space, Contemporary Math. 54 (1986), 17-33.
15. M. P. HEBLE, Approximation problems in analysis and probability North Holland, 1989
16. M.C. IRWIN, On the smoothness of the composition map, Quart. Jour. Math. 23 (1972), 113-133.
17. S. KRANTZ, Lipschitz spaces, smoothness of functions and approximation theory, Expo. Mat. 3 (1983), 193-260.
18. N. KURZWEIL On approximation in real Banach spaces, Studia Math. 14 (1954), 213-231
19. A. KATOK, G. KNIEPER, M. POLLICOTT, H. WEISS, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Inv. Math. 92 (1989), 581-597.
20. E. LANDAU, Einige ungleichungen für zweimal differentiarbare funktionen, Proc. London Math. Soc. 13 (1896), 43-58.
21. O. E. LANFORD III, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens in Lect. Notes in Math. 322, Springer-Verlag, 1973.
22. J. LANGER, D.A. SINGER, Diffeomorphisms of the circle and geodesic fields in Riemmann surfaces of genus one, Inv. Mat. 69 (1982), 229-242.
23. E. B. LEACH, J.H.M. WHITFIELD Differentiate functions and rough norms on Banach spaces, Proc. AMS, 33 (1972), 120-126
24. R. DE LA LLAVE, Cohomology equations over Anosov systems in non conmmutative groups, including groups of diffeomorphisms, Preprint.
25. R. DE LA LLAVE, J.M. MARCO, R. MORIYÓN, Canonical perturbation theories of Anosov diffeomorphisms and regularity results for the Livsic cohomology equation, Ann. of Math. 123 (1986), 537-611.
26. R. DE LA LLAVE, R. OBAYA, Decomposition theorems for some groups of diffeomorphisms, Trans. AMS. To appear
27. J. G. LLAVONA Approximation of continuously differentiable functions. North Holland, 1986
28. J.N. MATHER, Commutator of diffeomorphisms, Comment. Mat. Helv. 49 (1974), 512-528.
29. P. J. MORRISON Hamiltonian description of the ideal fluid Rev. Mod. Phys. 70 (1998) 467-522.
30. J. MOSER, A rapidly convergent iteration method and non-linear differential equations I and II, Ann. Sc. Norm. Pisa 20 (1966), 265-313 and 499-533.
31. J. MOSER, On a theorem of Anosov, Jour. Diff. Eg. 5 (1969), 411-440.
32. N. MOULIS, Approximation de fonctions differentiables sur certains espaces de Banach, Ann. Inst. Fourier 21, 4 (1971), 293-345.
33. S. M. NIKOLSKII, Approximation of functions of several variables and imbedding theorems Springer Verlag, New York, 1971.
34. V. NITICǍ, A. TÖRÖK, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. Jour. 79 (1995), 751-810.
35. R. S. PALAIS, Foundations of global non-linear analysis, Denjamin, New York, 1968.
36. D. RUELLE, Differentiation of SRB states, Deposited in mp_arc # 96-499.
37. S. SMALE, Differentiable dynamical systems, Bull. A.M.S. 73 (1967), 747-817.
38. E. STEIN, Singular integrals and differentiability properties of functions, Princeton University Press, 1970.
39. K. SUNDARESAN, Geometry and nonlinear analysis in Banach spaces, Pacific Journal of Math. 102, 2 (1982), 487-498.
40. M. VISHIK, Hydrodynamics in two dimensions, Preprint
41. E. ZEHNDER, Generalized implicit function theorems with applications to some small divisor problems I, Comm. Pure and Appl. Math. 28 (1975), 91-140.