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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

Mickaël D. Chekroun 1, and  Jean Roux 2,

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.
Keywords: Conjugacy problems, Koopman operator, functional equations and inequalities, spectral problems., homeomorphisms group, semimetric spaces.
Mathematics Subject Classification: 20E45, 37C15, 39B62, 39B72, 47A75, 47B33, 54A20, 54E15, 54E25, 57S0.
Citation: Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957
References:
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C. Aliprantis and K. Border, "Infinite Dimensional Analysis: A Hitchhiker's Guide,", Springer-Verlag, (2007).   Google Scholar

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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.  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.  doi: 10.1007/BF02921624.  Google Scholar

[8]

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[9]

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

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

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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.  Google Scholar

[12]

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

[13]

M. Chaperon, "Géométrie Différentielle et Singularités de Systèmes Dynamiques,", Astérisque, 138-139 (1986), 138.   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.  doi: 10.3934/dcds.2006.14.753.  Google Scholar

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C. C. Cowen and B. D. MacCluer, "Composition Operators on Spaces of Analytic Functions,", Studies in Advanced Mathematics, (1995).   Google Scholar

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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.  Google Scholar

[20]

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

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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.  Google Scholar

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

[25]

M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976).   Google Scholar

[26]

M. C. Irwin, "Smooth Dynamical Systems,", Reprint of the 1980 original, 17 (1980).  doi: 10.1142/9789812810120.  Google Scholar

[27]

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

[28]

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

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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).   Google Scholar

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J. L. Kelley, "General Topology,", Reprint of the 1955 edition [Van Nostrand, (1955).   Google Scholar

[31]

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

[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).   Google Scholar

[34]

M. Kuczma, "Functional Equations in a Single Variable,", Monografir Mat., 46 (1968).   Google Scholar

[35]

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

[36]

A. Lasota and M. C. Mackey, "Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics,", Second edition, 97 (1994).   Google Scholar

[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.  Google Scholar

[41]

M. Misiurewicz, Strange attractors for the Lozi mappings,, in, 357 (1980), 348.   Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[48]

E. Seneta, Functional equations and the Galton-Watson process,, Advances in Applied Probability, 1 (1969), 1.   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.  doi: 10.1006/jfan.1995.1002.  Google Scholar

[50]

J. H. Shapiro, Composition operators and Schröder's functional equation,, in, 213 (1998), 213.  doi: 10.1090/conm/213/02861.  Google Scholar

[51]

S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms,, in, (1963), 490.   Google Scholar

[52]

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

[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.  Google Scholar

[54]

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

[4]

V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advanced Series Nonlinear Dynamics, 16 (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.   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.  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.  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.   Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

M. Chaperon, "Géométrie Différentielle et Singularités de Systèmes Dynamiques,", Astérisque, 138-139 (1986), 138.   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.  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.  doi: 10.1007/s00020-003-1250-z.  Google Scholar

[16]

N. D. Cong, "Topological Dynamics of Random Dynamical Systems,", Oxford Mathematical Monographs, (1997).   Google Scholar

[17]

I. Cornfeld, S. Fomin and Ya. Sinaĭ, "Ergodic Theory,", Grundlehren der Mathematischen Wissenschaften, 245 (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, (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.  doi: 10.2307/1971334.  Google Scholar

[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.  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.  doi: 10.1023/A:1021726902801.  Google Scholar

[24]

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

[25]

M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976).   Google Scholar

[26]

M. C. Irwin, "Smooth Dynamical Systems,", Reprint of the 1980 original, 17 (1980).  doi: 10.1142/9789812810120.  Google Scholar

[27]

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

[28]

R. R. Kallman, Uniqueness results for homeomorphism groups,, Trans. Amer. Math. Soc., 295 (1986), 389.  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, 54 (1995).   Google Scholar

[30]

J. L. Kelley, "General Topology,", Reprint of the 1955 edition [Van Nostrand, (1955).   Google Scholar

[31]

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

[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).   Google Scholar

[34]

M. Kuczma, "Functional Equations in a Single Variable,", Monografir Mat., 46 (1968).   Google Scholar

[35]

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

[36]

A. Lasota and M. C. Mackey, "Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics,", Second edition, 97 (1994).   Google Scholar

[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.  Google Scholar

[41]

M. Misiurewicz, Strange attractors for the Lozi mappings,, in, 357 (1980), 348.   Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[48]

E. Seneta, Functional equations and the Galton-Watson process,, Advances in Applied Probability, 1 (1969), 1.   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.  doi: 10.1006/jfan.1995.1002.  Google Scholar

[50]

J. H. Shapiro, Composition operators and Schröder's functional equation,, in, 213 (1998), 213.  doi: 10.1090/conm/213/02861.  Google Scholar

[51]

S. Smale, Dynamical systems and the topological conjugacy problem for diffeomorphisms,, in, (1963), 490.   Google Scholar

[52]

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

[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.  Google Scholar

[54]

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

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