American Institute of Mathematical Sciences

Advanced
August  2013, 33(8): 3355-3363. doi: 10.3934/dcds.2013.33.3355

An alternative approach to generalised BV and the application to expanding interval maps

Oliver Butterley 1,

1. 

Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scienti ca, 00133 Roma, Italy

Received  April 2012 Revised  May 2012 Published  January 2013

We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise Hölder continuous and the partition on which the map is continuous may possess a countable number of elements. For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.
Keywords: spectral gap., transfer operator, Generalised bounded variation, expanding map.
Mathematics Subject Classification: 37E05, 37A2.
Citation: Oliver Butterley. An alternative approach to generalised BV and the application to expanding interval maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3355-3363. doi: 10.3934/dcds.2013.33.3355
References:
[1]

V. Baladi, "Positive Transfer Operators & Decay of Correlation,", 16 of Advanced Series in Nonlinear Dynamics, 16 (2000).  doi: 10.1142/9789812813633.  Google Scholar

[2]

J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction,", 223, 223 (1976).   Google Scholar

[3]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens,, Proc. Amer. Math. Soc., 118 (1993), 627.  doi: 10.2307/2160348.  Google Scholar

[4]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations,, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461.  doi: 10.1007/BF00532744.  Google Scholar

[5]

S. Luzzatto, I. Melbourne and F. Paccaut, The Lorenz attractor is mixing,, Comm. Math. Phys., 260 (2005), 393.  doi: 10.1007/s00220-005-1411-9.  Google Scholar

[6]

R. D. Nussbaum, The radius of the essential spectrum,, Duke Math. J., 37 (1970), 473.   Google Scholar

[7]

M. Pollicott, On the rate of mixing of Axiom A flows,, Invent. Math., 81 (1985), 413.  doi: 10.1007/BF01388579.  Google Scholar

[8]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps,, Discrete Contin. Dyn. Syst., 30 (2011), 917.  doi: 10.3934/dcds.2011.30.917.  Google Scholar

show all references

References:
[1]

V. Baladi, "Positive Transfer Operators & Decay of Correlation,", 16 of Advanced Series in Nonlinear Dynamics, 16 (2000).  doi: 10.1142/9789812813633.  Google Scholar

[2]

J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction,", 223, 223 (1976).   Google Scholar

[3]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens,, Proc. Amer. Math. Soc., 118 (1993), 627.  doi: 10.2307/2160348.  Google Scholar

[4]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations,, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461.  doi: 10.1007/BF00532744.  Google Scholar

[5]

S. Luzzatto, I. Melbourne and F. Paccaut, The Lorenz attractor is mixing,, Comm. Math. Phys., 260 (2005), 393.  doi: 10.1007/s00220-005-1411-9.  Google Scholar

[6]

R. D. Nussbaum, The radius of the essential spectrum,, Duke Math. J., 37 (1970), 473.   Google Scholar

[7]

M. Pollicott, On the rate of mixing of Axiom A flows,, Invent. Math., 81 (1985), 413.  doi: 10.1007/BF01388579.  Google Scholar

[8]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps,, Discrete Contin. Dyn. Syst., 30 (2011), 917.  doi: 10.3934/dcds.2011.30.917.  Google Scholar

[1]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[2]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[3]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[4]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[5]

Konstantinos Papafitsoros, Kristian Bredies. A study of the one dimensional total generalised variation regularisation problem. Inverse Problems & Imaging, 2015, 9 (2) : 511-550. doi: 10.3934/ipi.2015.9.511
[6]

Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
[7]

Chris Guiver. The generalised singular perturbation approximation for bounded real and positive real control systems. Mathematical Control & Related Fields, 2019, 9 (2) : 313-350. doi: 10.3934/mcrf.2019016
[8]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[9]

Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129
[10]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[11]

Soña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 53-69. doi: 10.3934/naco.2019005
[12]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235
[13]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241
[14]

Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43
[15]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[16]

Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[17]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835
[18]

Xing-Bin Pan. An eigenvalue variation problem of magnetic Schrödinger operator in three dimensions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 933-978. doi: 10.3934/dcds.2009.24.933
[19]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043
[20]

Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences