Skip to main content
SpringerLink
Log in

Square turning maps and their compactifications

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper we introduce some infinite rectangle exchange transformations which are based on the simultaneous turning of the squares within a sequence of square grids. We will show that such noncompact systems have higher dimensional dynamical compactifications. In good cases, these compactifications are polytope exchange transformations based on pairs of Euclidean lattices. In each dimension \(8m+4\) there is a \(4m+2\) dimensional family of them. Here \(m=0,1,2,\ldots \) We studied the case \(m=0\) in depth in Schwartz (The octagonal PETs, research monograph, 2012).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. Somewhat later on, we will find it more useful to set \(z_1=(1+i)/2\), to that the origin is the center of a square of the first grid, rather than a vertex of the grid.

  2. Note that the maps \(H(A_i)\) are rigid motions of the plane. They are not directly related to the square turning maps defined above.

  3. We often find it more convenient to work in \({\varvec{C}}^k\) rather than \({\varvec{R}}^{2k}\), even though sometimes we will make the identification of \({\varvec{C}}^k\) with \({\varvec{R}}^{2k}\).

  4. There is a certain redundancy, in the sense that the data \(\{\lambda s_1,\ldots ,\lambda s_n\}\) leads to a system which is conjugate to the system defined by \(\{s_1,\ldots ,s_n\}\). So, perhaps it is more accurate to say that there is a \(4m+1\) dimensional family of examples in dimension \(8m+4\).

  5. Not all double lattice PETs can be specified this way, but the ones here can be.

References

  1. Akiyama, S., Harriss, E.: Pentagonal Domain Exchange. Discret. Contin. Dyn. Syst. A 33(10), 4375–4400 (2013)

  2. Adler, R., Kitchens, B., Tresser, C.: Dynamics of non-ergodic piecewise affine maps of the torus. Ergod. Theory Dyn. Syst. 21(4), 959–999 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bedaride, N., Cassaigne, J.: Outer billiard outside regular polygons. J. Lond. Math. Soc. 84(2), 303–324 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buzzi, J.: Piecewise isometries have zero topological entropy (English summary). Ergod. Theory Dyn. Syst. 21(5), 1371–1377 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Goetz, A.: Piecewise Isometries—An Emerging Area of Dynamical Systems, preprint (2013)

  6. Gutkin, E., Haydn, N.: Topological entropy of generalized polygon exchanges. Bull. Am. Math. Soc. 32(1), 50–56 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goetz, A., Poggiaspalla, G.: Rotations by \(\pi /7\). Nonlinearity 17(5), 1787–1802 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gutkin, E., Haydn, N.: Topological entropy polygon exchange transformations and polygonal billiards. Ergod. Theory Dyn. Syst. 17(4), 849–867 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Haller, H.: Rectangle exchange transformations. Mon. Math. 91, 215–232 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hooper, W.P.: Renormalization of polygon exchange maps arising from corner percolation. Invent. Math. 191(2), 255–320 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Keane, M.: Non-ergodic interval exchange transformations. Isr. J. Math. 26, 188–96 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lowenstein, J.H., Koupsov, K.L., Vivaldi, F.: Recursive tiling and geometry of piecewise rotations by \(\pi /7\). Nonlinearity 17(2), 3710396 (2004)

    Article  MathSciNet  Google Scholar 

  13. Lowenstein, J.H.: Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling. Dyn. Syst. 22(1), 25–63 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lowenstein, J.H.: Pseudochaotic Kicked Oscillators. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  15. Rauzy, G.: Exchanges d’intervalles et transformations induites. Acta. Arith. 34, 315–328 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schwartz, R.E.: The Octagonal PETs, A.M.S. Mathematical Surveys and Monographs Number 197, 212 pp (2014)

  17. Schwartz, R.E.: Outer billiards on kites. Ann. Math. Stud. 171, 306 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Schwartz, R.E.: Outer Billiards, Quarter Turn Compositions, and Polytope Exchange Transformations, preprint (2010)

  19. Tabachnikov, S.: Billiards, Société Mathématique de France, “Panoramas et Syntheses”, vol. 1 (1995)

  20. Yoccoz, J.-C.: Continued fraction algorithms for interval exchange maps: an introduction. In: Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number Theory, Physics, and Geometry, vol. 1. Springer, Berlin, 4030437 (2006)

  21. Zorich, A.: Flat surfaces. In: Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number Theory, Physics, and Geometry, vol. 1. Springer, Berlin, 4030437 (2006)

Download references

Acknowledgements

I would like to thank Nicolas Bedaride, Pat Hooper, Injee Jeong, John Smillie, and Sergei Tabachnikov for interesting conversations about topics related to this work. I wrote this paper during my sabbatical at Oxford in 2012-20-13. I would especially like to thank All Souls College, Oxford, for providing a wonderful research environment. My sabbatical was funded from many sources. I would like to thank the National Science Foundation, All Souls College, the Oxford Maths Institute, the Simons Foundation, the Leverhulme Trust, the Chancellor’s Professorship, and Brown University for their support during this time period.

Author information

Authors and Affiliations

  1. Brown University, College Park, MD, USA

    Richard Evan Schwartz

Authors
  1. Richard Evan Schwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard Evan Schwartz.

Additional information

Supported by N.S.F. Research Grant DMS-0072607.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schwartz, R.E. Square turning maps and their compactifications. Geom Dedicata 192, 295–325 (2018). https://doi.org/10.1007/s10711-017-0246-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-017-0246-9

Keywords

Mathematics Subject Classification

Search

Navigation