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Natural extensions for piecewise affine maps via Hofbauer towers

Abstract

We use canonical Markov extensions (Hofbauer towers) to give an explicit construction of the natural extensions of various measure preserving endomorphisms, and present some applications to particular examples.

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Notes

  1. But not the generalised continued fraction maps of e.g. [33, 47] because there the Jacobian is not piecewise constant.

  2. Recall that the space \(X\) and also the levels \(D\) are compact, so \((\hat{\mu }_n|_D)_n\) has a weak accumulation point for each \(D\).

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Acknowledgments

The authors want to thank the referee for the useful remarks and suggested literature.

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Correspondence to Henk Bruin.

Additional information

Communicated by A. Constantin.

C. Kalle was supported by the research grant FWF S6913 and the NWO Veni grant 693.031.140. The research was partly supported by the EU FP6 Marie Curie Research Training Network CODY (MRTN 2006 035651).

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Bruin, H., Kalle, C. Natural extensions for piecewise affine maps via Hofbauer towers. Monatsh Math 175, 65–88 (2014). https://doi.org/10.1007/s00605-014-0644-0

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  • DOI: https://doi.org/10.1007/s00605-014-0644-0

Keywords

  • Natural extension
  • Piecewise affine maps
  • Hofbauer tower

Mathematics Subject Classification (2000)

  • 37A05
  • 37B10
  • 28A75