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April  2019, 39(4): 1779-1797. doi: 10.3934/dcds.2019077

Spectra of expanding maps on Besov spaces

Yushi Nakano 1, and  Shota Sakamoto 2,

1. 

Kitami Institute of Technology, Faculty of Engineering, 165 Koen-cho, Kitami, Hokkaido, 090-8507, Japan
2. 

Kyoto University, Graduate School of Human and Environmental Studies, Yoshida Nihonmatsu-cho, Sakyo-ku, Kyoto, 606-8501, Japan

Received  October 2017 Revised  September 2018 Published  January 2019

Fund Project: This work was partially supported by JSPS KAKENHI Grant Numbers 16J03963 and 17K05283.

A typical approach to analysing statistical properties of expanding maps is to show spectral gaps of associated transfer operators in adapted function spaces. The classical function spaces for this purpose are Hölder spaces and Sobolev spaces. Natural generalisations of these spaces are Besov spaces, on which we show a spectral gap of transfer operators.

Keywords: Transfer operator, Besov space, expanding map, Littlewood-Paley decomposition, quasi-compactness.
Mathematics Subject Classification: Primary: 37C30; Secondary: 37D20.
Citation: Yushi Nakano, Shota Sakamoto. Spectra of expanding maps on Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1779-1797. doi: 10.3934/dcds.2019077
References:
[1]

J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps, Stochastics and Dynamics, 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

M. Baillif and V. Baladi, Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case, Ergodic Theory Dyn. Syst., 25 (2005), 1437-1470.  doi: 10.1017/S014338570500012X.

[4]

V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, 2000. doi: 10.1142/9789812813633.

[5]

V. Baladi, The quest for the ultimate anisotropic Banach space, Journal of Statistical Physics, 166 (2017), 525-557.  doi: 10.1007/s10955-016-1663-0.

[6]

V. Baladi, Characteristic functions as bounded multipliers on anisotropic spaces, Proceedings of the American Mathematical Society, 146 (2018), 4405-4420.  doi: 10.1090/proc/14107.

[7]

V. Baladi, Correction to: The Quest for the Ultimate Anisotropic Banach Space, Journal of Statistical Physics, 170 (2018), 1242-1247.  doi: 10.1007/s10955-018-1976-2.

[8]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, 2018. doi: 10.1007/978-3-319-77661-3.

[9]

V. Baladi, A. Kondah and B. Schmitt, Random correlations for small perturbations of expanding maps, Random Comput. Dynam., 4 (1996), 179-204. 

[10]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, 57 (2007), 127-154.  doi: 10.5802/aif.2253.

[11]

V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in: K. Burns, D. Dolgopyat and Y. Pesin (eds.), Geometric and probabilistic structures in dynamics, Contemp. Math., Amer. Math. Soc., 469 (2008), 29–68. doi: 10.1090/conm/469.

[12]

V. Baladi and M. Tsujii, Spectra of differential hyperbolic maps, in: S. Albeverio, M. Marcolli, S. Paycha and J. Plazas (eds.), Traces in number theory, geometry and quantum fields, Aspects Math., Friedr. Vieweg, 38 (2008), 1–21.

[13]

V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385.  doi: 10.1007/BF02098487.

[14]

V. Baladi and L.-S. Young, Erratum: "On the spectra of randomly perturbed expanding maps", Comm. Math. Phys., 166 (1994), 219-220. 

[15]

F. Faure, Semiclassical origin of the spectral gap for transfer operators of a partially expanding map, Nonlinearity, 24 (2011), 1473-1498.  doi: 10.1088/0951-7715/24/5/005.

[16]

S. Gouëzel, Limit theorems in dynamical systems using the spectral method, in: D. Dolgopyat, Y. Pesin, M. Pollicott and L. Stoyanov (eds.), Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Sympos. Pure Math., Amer. Math. Soc., 89 (2015), 161–193. doi: 10.1090/pspum/089/01487.

[17]

V. M. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces, Ergodic Theory and Dynamical Systems, 23 (2003), 175-191.  doi: 10.1017/S0143385702000962.

[18]

H. Hennion, Sur un théoreme spectral et son application aux noyaux lipchitziens, Proceedings of the American Mathematical Society, 118 (1993), 627-634.  doi: 10.2307/2160348.

[19] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.
[20]

R. Mané, Ergodic Theory and Differentiable Dynamics, vol. 8, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.

[21]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, vol. 371, Cambridge University Press, 2010. doi: 10.1017/CBO9781139193184.

[22]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley, 1978.

[23]

D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys., 125 (1989), 239-262.  doi: 10.1007/BF01217908.

[24]

D. Ruelle, Une extension de la théorie de Fredholm, Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 309 (1989), 309–310.

[25]

D. Ruelle, An extension of the theory of Fredholm determinants, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 175–193.

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, vol. 2, Princeton university press, 1970.

[27]

M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, vol. 100 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.

[28]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Continuous Dynamical Systems-A, 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917.

[29]

H. Triebel, Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation, Arkiv för Matematik, 11 (1973), 13–64. doi: 10.1007/BF02388506.

[30]

H. Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Arkiv för Matematik, 24 (1986), 299–337. doi: 10.1007/BF02384402.

[31]

M. Viana, Lectures on Lyapunov exponents, vol. 145, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.

[32]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.

show all references

References:
[1]

J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps, Stochastics and Dynamics, 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

M. Baillif and V. Baladi, Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case, Ergodic Theory Dyn. Syst., 25 (2005), 1437-1470.  doi: 10.1017/S014338570500012X.

[4]

V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, 2000. doi: 10.1142/9789812813633.

[5]

V. Baladi, The quest for the ultimate anisotropic Banach space, Journal of Statistical Physics, 166 (2017), 525-557.  doi: 10.1007/s10955-016-1663-0.

[6]

V. Baladi, Characteristic functions as bounded multipliers on anisotropic spaces, Proceedings of the American Mathematical Society, 146 (2018), 4405-4420.  doi: 10.1090/proc/14107.

[7]

V. Baladi, Correction to: The Quest for the Ultimate Anisotropic Banach Space, Journal of Statistical Physics, 170 (2018), 1242-1247.  doi: 10.1007/s10955-018-1976-2.

[8]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, 2018. doi: 10.1007/978-3-319-77661-3.

[9]

V. Baladi, A. Kondah and B. Schmitt, Random correlations for small perturbations of expanding maps, Random Comput. Dynam., 4 (1996), 179-204. 

[10]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, 57 (2007), 127-154.  doi: 10.5802/aif.2253.

[11]

V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in: K. Burns, D. Dolgopyat and Y. Pesin (eds.), Geometric and probabilistic structures in dynamics, Contemp. Math., Amer. Math. Soc., 469 (2008), 29–68. doi: 10.1090/conm/469.

[12]

V. Baladi and M. Tsujii, Spectra of differential hyperbolic maps, in: S. Albeverio, M. Marcolli, S. Paycha and J. Plazas (eds.), Traces in number theory, geometry and quantum fields, Aspects Math., Friedr. Vieweg, 38 (2008), 1–21.

[13]

V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385.  doi: 10.1007/BF02098487.

[14]

V. Baladi and L.-S. Young, Erratum: "On the spectra of randomly perturbed expanding maps", Comm. Math. Phys., 166 (1994), 219-220. 

[15]

F. Faure, Semiclassical origin of the spectral gap for transfer operators of a partially expanding map, Nonlinearity, 24 (2011), 1473-1498.  doi: 10.1088/0951-7715/24/5/005.

[16]

S. Gouëzel, Limit theorems in dynamical systems using the spectral method, in: D. Dolgopyat, Y. Pesin, M. Pollicott and L. Stoyanov (eds.), Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Sympos. Pure Math., Amer. Math. Soc., 89 (2015), 161–193. doi: 10.1090/pspum/089/01487.

[17]

V. M. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces, Ergodic Theory and Dynamical Systems, 23 (2003), 175-191.  doi: 10.1017/S0143385702000962.

[18]

H. Hennion, Sur un théoreme spectral et son application aux noyaux lipchitziens, Proceedings of the American Mathematical Society, 118 (1993), 627-634.  doi: 10.2307/2160348.

[19] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.
[20]

R. Mané, Ergodic Theory and Differentiable Dynamics, vol. 8, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.

[21]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, vol. 371, Cambridge University Press, 2010. doi: 10.1017/CBO9781139193184.

[22]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley, 1978.

[23]

D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys., 125 (1989), 239-262.  doi: 10.1007/BF01217908.

[24]

D. Ruelle, Une extension de la théorie de Fredholm, Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 309 (1989), 309–310.

[25]

D. Ruelle, An extension of the theory of Fredholm determinants, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 175–193.

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, vol. 2, Princeton university press, 1970.

[27]

M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, vol. 100 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.

[28]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Continuous Dynamical Systems-A, 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917.

[29]

H. Triebel, Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation, Arkiv för Matematik, 11 (1973), 13–64. doi: 10.1007/BF02388506.

[30]

H. Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Arkiv för Matematik, 24 (1986), 299–337. doi: 10.1007/BF02384402.

[31]

M. Viana, Lectures on Lyapunov exponents, vol. 145, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.

[32]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.

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