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Invariant measures with bounded variation densities for piecewise area preserving maps

Published online by Cambridge University Press:  14 February 2012

YIWEI ZHANG  and
CONGPING LIN
YIWEI ZHANG
Affiliation:
Mathematics Research Institute, University of Exeter, Exeter EX4 4QF, UK (email: yz297@exeter.ac.uk, cl336@exeter.ac.uk)
CONGPING LIN
Affiliation:
Mathematics Research Institute, University of Exeter, Exeter EX4 4QF, UK (email: yz297@exeter.ac.uk, cl336@exeter.ac.uk)
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Abstract

We investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 624 - 642
Copyright
©2012 Cambridge University Press

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References

[1]Ashwin, P., Fu, X. C. and Terry, J. R.. Riddling and invariance for discontinuous parabolic maps of torus. Nonlinearity 13 (2002), 818–835.Google Scholar
[2]Baladi, V. and Gouëzel, S.. Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1453–1481.Google Scholar
[3]Boyarsky, A. and Góra, P.. Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. Birkhäuser, Boston, MA, 1997.Google Scholar
[4]Bruin, H. and Troubetzkoy, S.. The Gauss map on a class of interval translation mappings. Israel J. Math. 137 (2003), 125–148.CrossRefGoogle Scholar
[5]Buzzi, J.. No or infinitely many a.c.i.p. for piecewise expanding C r maps in higher dimensions. Comm. Math. Phys. 222 (2001), 495–501.CrossRefGoogle Scholar
[6]Coffey, J.. Some remarks concerning an example of a minimal non-unique ergodic interval exchange transformation. Math. Z. 199 (1988), 577–580.Google Scholar
[7]Chaika, J.. Hausdorff dimension for ergodic measures of interval exchange transformations. J. Mod. Dyn. 2 (2008), 457–464.CrossRefGoogle Scholar
[8]Driver, B. K.. Analysis Tools with Applications. Springer, Berlin, 2003.Google Scholar
[9]Evans, L. C. and Gariepy, R. F.. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL, 1992.Google Scholar
[10]Fayad, B. and Katok, A.. Constructions in elliptic dynamics. Ergod. Th. & Dynam. Sys. 24 (2004), 1477–1520.Google Scholar
[11]Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Hoboken, NJ, 2003.Google Scholar
[12]Fu, X. C., Chen, F. Y. and Zhao, X. H.. Dynamical properties of 2-torus parabolic maps. Nonlinear Dynam. 50 (2007), 539–549.Google Scholar
[13]Goetz, A.. Piecewise isometries—an emerging area of dynamical systems. Fractals in Graz 2001: Analysis, Dynamics, Geometry, Stochastics. Eds. Grabner, P. and Woess, W.. Birkhäuser, Basel, 2002, pp. 135–144.Google Scholar
[14]Goetz, A.. Stability of piecewise rotations and affine maps. Nonlinearity 14 (2001), 205–219.Google Scholar
[15]Jost, J.. Postmodern Analysis. Springer, Berlin, 2005.Google Scholar
[16]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, London, 1995.Google Scholar
[17]Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
[18]Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26 (1977), 188–196.Google Scholar
[19]Keane, M. and Rauzy, G.. Stricte ergodicité des échanges d’intervalles. Math. Z. 174 (1980), 203–212.Google Scholar
[20]Keller, G.. Coupled map lattices via transfer operators on functions of bounded variation. Stochastic and Spatial Structures of Dynamical Systems (Amsterdam, 1995) (Koninklijke Nederlandse Akademie van Wetenschappen. Verhandelingen, Afd. Natuurkunde. Eerste Reeks, 45). Eds. van Strien, S. J. and Verduyn Lunel, S. M.. North-Holland, Amsterdam, 1996, pp. 71–80.Google Scholar
[21]Keynes, H. B. and Newton, D.. A minimal, non-uniquely ergodic interval exchange transformation. Math. Z. 148 (1976), 101–105.Google Scholar
[22]MacKay, R. S.. Renormalisation in Area-preserving Maps. World Scientific, Singapore, 1993.Google Scholar
[23]Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. 115 (1982), 169–200.Google Scholar
[24]Peris, A.. Transitivity, dense orbit and discontinuous functions. Bull. Belg. Math. Soc. Simon Stevin 6 (1999), 391–394.Google Scholar
[25]Saussol, B.. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116 (2000), 223–248.Google Scholar
[26]Tsujii, M.. Piecewise expanding maps on the plane with singular ergodic properties. Ergod. Th. & Dynam. Sys. 20 (2000), 1851–1857.Google Scholar
[27]Veech, W.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242.Google Scholar
[28]Viana, M.. Stochastic Dynamics of Deterministic Systems (Brazilian Mathematics Colloquium, 21). IMPA, Rio de Janeiro, 1997.Google Scholar
[29]Vol’pert, A. I. and Hudjaev, S. I.. Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus Nijhoff, Dordrecht, 1985.Google Scholar
[30]Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.Google Scholar