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

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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 Scientica, 00133 Roma, Italy

Received April 2012 Revised May 2012 Published January 2013

Abstract
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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 and Continuous Dynamical Systems, 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, World Scientific, Singapore, 2000. doi: 10.1142/9789812813633.
[2]	J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction," 223, Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, 1976.
[3]	H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348.
[4]	G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478. doi: 10.1007/BF00532744.
[5]	S. Luzzatto, I. Melbourne and F. Paccaut, The Lorenz attractor is mixing, Comm. Math. Phys., 260 (2005), 393-401. doi: 10.1007/s00220-005-1411-9.
[6]	R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478.
[7]	M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math., 81 (1985), 413-426. doi: 10.1007/BF01388579.
[8]	D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917.

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References:

[1]	V. Baladi, "Positive Transfer Operators & Decay of Correlation," 16 of Advanced Series in Nonlinear Dynamics, World Scientific, Singapore, 2000. doi: 10.1142/9789812813633.
[2]	J. Bergh and J. Löfström, "Interpolation Spaces, An Introduction," 223, Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, 1976.
[3]	H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348.
[4]	G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478. doi: 10.1007/BF00532744.
[5]	S. Luzzatto, I. Melbourne and F. Paccaut, The Lorenz attractor is mixing, Comm. Math. Phys., 260 (2005), 393-401. doi: 10.1007/s00220-005-1411-9.
[6]	R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478.
[7]	M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math., 81 (1985), 413-426. doi: 10.1007/BF01388579.
[8]	D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917.

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