An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions

Previous Article
Thermodynamic invariants of Anosov flows and rigidity
DCDS Home
This Issue
Next Article
On a question of Katok in one-dimensional case

November 2009, 24(4): 1205-1208. doi: 10.3934/dcds.2009.24.1205

An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions

1.	IRMAR, Université de Rennes 1, 35042 Rennes, France

Received July 2008 Revised December 2008 Published May 2009

Abstract
Related Papers

We construct a uniformly expanding map of the interval, preserving Lebesgue measure, such that the corresponding transfer operator admits a spectral gap on the space of Lipschitz functions, but does not act continuously on the space of bounded variation functions.

Keywords: bounded variation functions., spectral gap, Lipschitz functions, Transfer operator.

Mathematics Subject Classification: 37E05, 37D2.

Citation: 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

[1]	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
[2]	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
[3]	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
[4]	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
[5]	Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33
[6]	Jian Liu, Sihem Mesnager, Lusheng Chen. Variation on correlation immune Boolean and vectorial functions. Advances in Mathematics of Communications, 2016, 10 (4) : 895-919. doi: 10.3934/amc.2016048
[7]	Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[8]	Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[9]	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
[10]	Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157
[11]	Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[12]	Minghua Li, Chunrong Chen, Shengjie Li. Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019001
[13]	Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843
[14]	Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems & Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565
[15]	Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39
[16]	Ken Abe. Some uniqueness result of the Stokes flow in a half space in a space of bounded functions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 887-900. doi: 10.3934/dcdss.2014.7.887
[17]	Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[18]	Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control & Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365
[19]	Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037
[20]	Sangtae Jeong, Chunlan Li. Measure-preservation criteria for a certain class of 1-lipschitz functions on Z_p in mahler's expansion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3787-3804. doi: 10.3934/dcds.2017160

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences