September  2013, 33(9): 3957-3980. doi: 10.3934/dcds.2013.33.3957

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

Received  October 2011 Revised  February 2013 Published  March 2013

This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.
Citation: Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957
References:
[1]

N. H. Abel, Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable, in "Oeuvres Complètes," 2 (1839), 246-248; "Oeuvres Complètes," 2, Christiania, Oslo, (1881), 36-39. Google Scholar

[2]

C. Aliprantis and K. Border, "Infinite Dimensional Analysis: A Hitchhiker's Guide," Springer-Verlag, New York, 2007. Google Scholar

[3]

R. Arens, Topologies for homeomorphism groups, Amer J. Math., 68 (1946), 593-610.  Google Scholar

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V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

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J. Banaś, A. Hajnosz and S. Wędrychowicz, On existence and asymptotic behavior of solutions of some functional equations, Funkcialaj Ekvacioj, 25 (1982), 257-267.  Google Scholar

[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), 755-778. doi: 10.2307/2001010.  Google Scholar

[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), 613-649. doi: 10.1007/BF02921624.  Google Scholar

[8]

G. Belitskii and Yu. Lyubich, The Abel equation and total solvability of linear functional equations, Studia Mathematica, 127 (1998), 81-97.  Google Scholar

[9]

G. Belitskii and Yu. Lyubich, The real-analytic solutions of the Abel functional equation, Studia Mathematica, 134 (1999), 135-141.  Google Scholar

[10]

G. Belitskii and V. Tkachenko, Functional equations in real-analytic functions, Studia Mathematica, 143 (2000), 153-174.  Google Scholar

[11]

P. S. Bourdon and J. H. Shapiro, Mean growth of Koenigs eigenfunctions, J. Amer. Math. Soc., 10 (1997), 299-325. doi: 10.1090/S0894-0347-97-00224-5.  Google Scholar

[12]

J. Caugran and H. J. Schwartz, Spectra of compact composition operators, Proc. Amer. Math. Soc., 51 (1975), 127-130.  Google Scholar

[13]

M. Chaperon, "Géométrie Différentielle et Singularités de Systèmes Dynamiques," Astérisque, 138-139, (1986), 440 pp.  Google Scholar

[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), 753-782. doi: 10.3934/dcds.2006.14.753.  Google Scholar

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D. D. Clahane, Spectra of compact composition operators over bounded symmetric domains, Integr. Equ. Oper. Theory, 51 (2005), 41-56. doi: 10.1007/s00020-003-1250-z.  Google Scholar

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N. D. Cong, "Topological Dynamics of Random Dynamical Systems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1997.  Google Scholar

[17]

I. Cornfeld, S. Fomin and Ya. Sinaĭ, "Ergodic Theory," Grundlehren der Mathematischen Wissenschaften, 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[18]

C. C. Cowen and B. D. MacCluer, "Composition Operators on Spaces of Analytic Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.  Google Scholar

[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), 537-611. doi: 10.2307/1971334.  Google Scholar

[20]

M. A. Denjoy, Sur l'itération de fonctions analytiques, C. R. Acad. Sci. Paris, 182 (1926), 255-257. Google Scholar

[21]

J. Dieudonné, "Éléments d'Analyse," Tome 1, Gauthiers-Villars, Paris, 1968. Google Scholar

[22]

J. Ding, The point spectrum of Perron-Frobenius and Koopman operators, Proc. Amer. Math. Soc., 126 (1998), 1355-1361. doi: 10.1090/S0002-9939-98-04188-4.  Google Scholar

[23]

G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems, 5 (1999), 173-226. doi: 10.1023/A:1021726902801.  Google Scholar

[24]

R. P. Gosselin, A maximal theorem for subadditive functions, Acta Mathematica, 112 (1964), 163-180.  Google Scholar

[25]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[26]

M. C. Irwin, "Smooth Dynamical Systems," Reprint of the 1980 original, With a foreword by R. S. MacKay, Advances Series in Nonlinear Dynamics, 17, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812810120.  Google Scholar

[27]

G. Julia, Sur une classe d'équations fonctionnelles, Annales Sci. de l'École Norm. Supérieure, Série 3, 40 (1923), 97-150.  Google Scholar

[28]

R. R. Kallman, Uniqueness results for homeomorphism groups, Trans. Amer. Math. Soc., 295 (1986), 389-396. doi: 10.2307/2000162.  Google Scholar

[29]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Anatole Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[30]

J. L. Kelley, "General Topology," Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975.  Google Scholar

[31]

G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Annales de l'École Normale Supérieure, 1 (1884), 3-41.  Google Scholar

[32]

B. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proc. Nat. Acad. Sci. USA, 18 (1932), 255-266. Google Scholar

[33]

J. Kotus, M. Krych and Z. Nitecki, Global structural stability of flows on open surfaces, Mem. Amer. Math. Soc., 37 (1982), v+108 pp.  Google Scholar

[34]

M. Kuczma, "Functional Equations in a Single Variable," Monografir Mat., 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968.  Google Scholar

[35]

M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations," Encyclopedia of Mathematics and its Applications, 32, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1017/CBO9781139086639.  Google Scholar

[36]

A. Lasota and M. C. Mackey, "Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics," Second edition, Applied Mathematical Sciences, 97, Springer-Verlag, New York, 1994.  Google Scholar

[37]

A. Livshitz, Homology properties of $Y$-systems, Math. Notes USSR Acad. Sci., 10 (1971), 758-763. Google Scholar

[38]

A. Livshitz, Cohomology of dynamical systems, Math. USSR-Izv, 6 (1972), 1278-1301. Google Scholar

[39]

R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon, J. Phys. (Paris), 39 (1978), 69-70. Google Scholar

[40]

I. Mezić and A. Banaszuk, Comparison of systems with complex behaviour, Physica D, 197 (2004), 101-133. doi: 10.1016/j.physd.2004.06.015.  Google Scholar

[41]

M. Misiurewicz, Strange attractors for the Lozi mappings, in "Nonlinear Dynamics" (Internat. Conf., New York, 1979), Ann. New York Acad. Sci., 357, New York Acad. Sci., New York, (1980), 348-358.  Google Scholar

[42]

J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001.  Google Scholar

[43]

R. A. Rosenbaum, Sub-additive functions, Duke Math. J., 17 (1950), 227-247.  Google Scholar

[44]

J. Ren and X. Zhang, Topologies on homeomorphism spaces of certain metric spaces, J. Math. Anal. Appl., 316 (2006), 32-36. doi: 10.1016/j.jmaa.2005.05.019.  Google Scholar

[45]

R. Roussarie and J. Roux, "Des Équations Différentielles aux Systèmes Dynamiques," Tomes I et II, EDP Sciences, 2012. Google Scholar

[46]

H. H. Schaefer, "Topological Vector Spaces," Second edition, Graduate Texts in Mathematics, 3, Springer-Verlag, 1999. Google Scholar

[47]

E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann., 2 (1870), 317-365. doi: 10.1007/BF01444024.  Google Scholar

[48]

E. Seneta, Functional equations and the Galton-Watson process, Advances in Applied Probability, 1 (1969), 1-42.  Google Scholar

[49]

J. H. Shapiro, W. Smith and D. A. Stegenga, Geometric models and compactness of composition operators, J. Functional Analysis, 127 (1995), 21-62. doi: 10.1006/jfan.1995.1002.  Google Scholar

[50]

J. H. Shapiro, Composition operators and Schröder's functional equation, in "Studies on Composition Operators" (Laramie, WY, 1996), Contemporary Mathematics, 213, Amer. Math. Soc., Providence, RI, (1998), 213-228. doi: 10.1090/conm/213/02861.  Google Scholar

[51]

S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms, in "Proceedings of the International Congress of Mathematicians" (ed. V. Stenström) (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, (1963), 490-496.  Google Scholar

[52]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  Google Scholar

[53]

J. Walorski, On the continuous smooth solutions of the Schröder equation in normed spaces, Integr. Equ. Oper. Theory, 60 (2008), 597-600. doi: 10.1007/s00020-007-1550-9.  Google Scholar

[54]

J.-C. Yoccoz, Théorème de Siegel, nombre de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88.  Google Scholar

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 "Oeuvres Complètes," 2 (1839), 246-248; "Oeuvres Complètes," 2, Christiania, Oslo, (1881), 36-39. Google Scholar

[2]

C. Aliprantis and K. Border, "Infinite Dimensional Analysis: A Hitchhiker's Guide," Springer-Verlag, New York, 2007. Google Scholar

[3]

R. Arens, Topologies for homeomorphism groups, Amer J. Math., 68 (1946), 593-610.  Google Scholar

[4]

V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[5]

J. Banaś, A. Hajnosz and S. Wędrychowicz, On existence and asymptotic behavior of solutions of some functional equations, Funkcialaj Ekvacioj, 25 (1982), 257-267.  Google Scholar

[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), 755-778. doi: 10.2307/2001010.  Google Scholar

[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), 613-649. doi: 10.1007/BF02921624.  Google Scholar

[8]

G. Belitskii and Yu. Lyubich, The Abel equation and total solvability of linear functional equations, Studia Mathematica, 127 (1998), 81-97.  Google Scholar

[9]

G. Belitskii and Yu. Lyubich, The real-analytic solutions of the Abel functional equation, Studia Mathematica, 134 (1999), 135-141.  Google Scholar

[10]

G. Belitskii and V. Tkachenko, Functional equations in real-analytic functions, Studia Mathematica, 143 (2000), 153-174.  Google Scholar

[11]

P. S. Bourdon and J. H. Shapiro, Mean growth of Koenigs eigenfunctions, J. Amer. Math. Soc., 10 (1997), 299-325. doi: 10.1090/S0894-0347-97-00224-5.  Google Scholar

[12]

J. Caugran and H. J. Schwartz, Spectra of compact composition operators, Proc. Amer. Math. Soc., 51 (1975), 127-130.  Google Scholar

[13]

M. Chaperon, "Géométrie Différentielle et Singularités de Systèmes Dynamiques," Astérisque, 138-139, (1986), 440 pp.  Google Scholar

[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), 753-782. doi: 10.3934/dcds.2006.14.753.  Google Scholar

[15]

D. D. Clahane, Spectra of compact composition operators over bounded symmetric domains, Integr. Equ. Oper. Theory, 51 (2005), 41-56. doi: 10.1007/s00020-003-1250-z.  Google Scholar

[16]

N. D. Cong, "Topological Dynamics of Random Dynamical Systems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1997.  Google Scholar

[17]

I. Cornfeld, S. Fomin and Ya. Sinaĭ, "Ergodic Theory," Grundlehren der Mathematischen Wissenschaften, 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[18]

C. C. Cowen and B. D. MacCluer, "Composition Operators on Spaces of Analytic Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.  Google Scholar

[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), 537-611. doi: 10.2307/1971334.  Google Scholar

[20]

M. A. Denjoy, Sur l'itération de fonctions analytiques, C. R. Acad. Sci. Paris, 182 (1926), 255-257. Google Scholar

[21]

J. Dieudonné, "Éléments d'Analyse," Tome 1, Gauthiers-Villars, Paris, 1968. Google Scholar

[22]

J. Ding, The point spectrum of Perron-Frobenius and Koopman operators, Proc. Amer. Math. Soc., 126 (1998), 1355-1361. doi: 10.1090/S0002-9939-98-04188-4.  Google Scholar

[23]

G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems, 5 (1999), 173-226. doi: 10.1023/A:1021726902801.  Google Scholar

[24]

R. P. Gosselin, A maximal theorem for subadditive functions, Acta Mathematica, 112 (1964), 163-180.  Google Scholar

[25]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[26]

M. C. Irwin, "Smooth Dynamical Systems," Reprint of the 1980 original, With a foreword by R. S. MacKay, Advances Series in Nonlinear Dynamics, 17, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812810120.  Google Scholar

[27]

G. Julia, Sur une classe d'équations fonctionnelles, Annales Sci. de l'École Norm. Supérieure, Série 3, 40 (1923), 97-150.  Google Scholar

[28]

R. R. Kallman, Uniqueness results for homeomorphism groups, Trans. Amer. Math. Soc., 295 (1986), 389-396. doi: 10.2307/2000162.  Google Scholar

[29]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Anatole Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[30]

J. L. Kelley, "General Topology," Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975.  Google Scholar

[31]

G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Annales de l'École Normale Supérieure, 1 (1884), 3-41.  Google Scholar

[32]

B. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proc. Nat. Acad. Sci. USA, 18 (1932), 255-266. Google Scholar

[33]

J. Kotus, M. Krych and Z. Nitecki, Global structural stability of flows on open surfaces, Mem. Amer. Math. Soc., 37 (1982), v+108 pp.  Google Scholar

[34]

M. Kuczma, "Functional Equations in a Single Variable," Monografir Mat., 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968.  Google Scholar

[35]

M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations," Encyclopedia of Mathematics and its Applications, 32, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1017/CBO9781139086639.  Google Scholar

[36]

A. Lasota and M. C. Mackey, "Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics," Second edition, Applied Mathematical Sciences, 97, Springer-Verlag, New York, 1994.  Google Scholar

[37]

A. Livshitz, Homology properties of $Y$-systems, Math. Notes USSR Acad. Sci., 10 (1971), 758-763. Google Scholar

[38]

A. Livshitz, Cohomology of dynamical systems, Math. USSR-Izv, 6 (1972), 1278-1301. Google Scholar

[39]

R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon, J. Phys. (Paris), 39 (1978), 69-70. Google Scholar

[40]

I. Mezić and A. Banaszuk, Comparison of systems with complex behaviour, Physica D, 197 (2004), 101-133. doi: 10.1016/j.physd.2004.06.015.  Google Scholar

[41]

M. Misiurewicz, Strange attractors for the Lozi mappings, in "Nonlinear Dynamics" (Internat. Conf., New York, 1979), Ann. New York Acad. Sci., 357, New York Acad. Sci., New York, (1980), 348-358.  Google Scholar

[42]

J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 22 (2005), 485-507. doi: 10.1016/j.anihpc.2005.01.001.  Google Scholar

[43]

R. A. Rosenbaum, Sub-additive functions, Duke Math. J., 17 (1950), 227-247.  Google Scholar

[44]

J. Ren and X. Zhang, Topologies on homeomorphism spaces of certain metric spaces, J. Math. Anal. Appl., 316 (2006), 32-36. doi: 10.1016/j.jmaa.2005.05.019.  Google Scholar

[45]

R. Roussarie and J. Roux, "Des Équations Différentielles aux Systèmes Dynamiques," Tomes I et II, EDP Sciences, 2012. Google Scholar

[46]

H. H. Schaefer, "Topological Vector Spaces," Second edition, Graduate Texts in Mathematics, 3, Springer-Verlag, 1999. Google Scholar

[47]

E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann., 2 (1870), 317-365. doi: 10.1007/BF01444024.  Google Scholar

[48]

E. Seneta, Functional equations and the Galton-Watson process, Advances in Applied Probability, 1 (1969), 1-42.  Google Scholar

[49]

J. H. Shapiro, W. Smith and D. A. Stegenga, Geometric models and compactness of composition operators, J. Functional Analysis, 127 (1995), 21-62. doi: 10.1006/jfan.1995.1002.  Google Scholar

[50]

J. H. Shapiro, Composition operators and Schröder's functional equation, in "Studies on Composition Operators" (Laramie, WY, 1996), Contemporary Mathematics, 213, Amer. Math. Soc., Providence, RI, (1998), 213-228. doi: 10.1090/conm/213/02861.  Google Scholar

[51]

S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms, in "Proceedings of the International Congress of Mathematicians" (ed. V. Stenström) (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, (1963), 490-496.  Google Scholar

[52]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  Google Scholar

[53]

J. Walorski, On the continuous smooth solutions of the Schröder equation in normed spaces, Integr. Equ. Oper. Theory, 60 (2008), 597-600. doi: 10.1007/s00020-007-1550-9.  Google Scholar

[54]

J.-C. Yoccoz, Théorème de Siegel, nombre de Bruno et polynômes quadratiques, Astérisque, 231 (1995), 3-88.  Google Scholar

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