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May  2014, 34(5): 2307-2314. doi: 10.3934/dcds.2014.34.2307

Almost every interval translation map of three intervals is finite type

Denis Volk 1,

1. 

Kungliga Tekniska Hogskolan (Royal Institute of Technology), Department of mathematics, SE-100 44, Stockholm, Sweden

Received  November 2012 Revised  August 2013 Published  October 2013

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.
Keywords: color rotations., interval exchange maps, double rotations, Dynamical systems, interval translation maps, attractors.
Mathematics Subject Classification: Primary: 37C05, 37C20, 37C70, 37D20, 37D4.
Citation: Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
References:
[1]

M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652.  Google Scholar

[2]

H. Bruin, Renormalization in a class of interval translation maps of $d$ branches, Dyn. Syst., 22 (2007), 11-24. doi: 10.1080/14689360601028084.  Google Scholar

[3]

H. Bruin and G. Clack, Inducing and unique ergodicity of double rotations, Discrete Contin. Dyn. Syst., 32 (2012), 4133-4147. doi: 10.3934/dcds.2012.32.4133.  Google Scholar

[4]

H. Bruin and S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Israel J. Math., 137 (2003), 125-148. doi: 10.1007/BF02785958.  Google Scholar

[5]

J. Buzzi, Piecewise isometries have zero topological entropy, Ergodic Theory Dynam. Systems, 21 (2001), 1371-1377. doi: 10.1017/S0143385701001651.  Google Scholar

[6]

J. Buzzi and P. Hubert, Piecewise monotone maps without periodic points: Rigidity, measures and complexity, Ergodic Theory Dynam. Systems, 24 (2004), 383-405. doi: 10.1017/S0143385703000488.  Google Scholar

[7]

A. Goetz, Sofic subshifts and piecewise isometric systems, Ergodic Theory Dynam. Systems, 19 (1999), 1485-1501. doi: 10.1017/S0143385799151964.  Google Scholar

[8]

A. Goetz, Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478.  Google Scholar

[9]

A. Goetz, Stability of piecewise rotations and affine maps, Nonlinearity, 14 (2001), 205-219. doi: 10.1088/0951-7715/14/2/302.  Google Scholar

[10]

J. Schmeling and S. Troubetzkoy, Interval Translation Mappings, In Dynamical systems (Luminy-Marseille, 1998), 291-302. World Sci. Publ., River Edge, NJ, 2000.  Google Scholar

[11]

H. Suzuki, S. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Syst., 13 (2005), 515-532. doi: 10.3934/dcds.2005.13.515.  Google Scholar

show all references

References:
[1]

M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652.  Google Scholar

[2]

H. Bruin, Renormalization in a class of interval translation maps of $d$ branches, Dyn. Syst., 22 (2007), 11-24. doi: 10.1080/14689360601028084.  Google Scholar

[3]

H. Bruin and G. Clack, Inducing and unique ergodicity of double rotations, Discrete Contin. Dyn. Syst., 32 (2012), 4133-4147. doi: 10.3934/dcds.2012.32.4133.  Google Scholar

[4]

H. Bruin and S. Troubetzkoy, The Gauss map on a class of interval translation mappings, Israel J. Math., 137 (2003), 125-148. doi: 10.1007/BF02785958.  Google Scholar

[5]

J. Buzzi, Piecewise isometries have zero topological entropy, Ergodic Theory Dynam. Systems, 21 (2001), 1371-1377. doi: 10.1017/S0143385701001651.  Google Scholar

[6]

J. Buzzi and P. Hubert, Piecewise monotone maps without periodic points: Rigidity, measures and complexity, Ergodic Theory Dynam. Systems, 24 (2004), 383-405. doi: 10.1017/S0143385703000488.  Google Scholar

[7]

A. Goetz, Sofic subshifts and piecewise isometric systems, Ergodic Theory Dynam. Systems, 19 (1999), 1485-1501. doi: 10.1017/S0143385799151964.  Google Scholar

[8]

A. Goetz, Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478.  Google Scholar

[9]

A. Goetz, Stability of piecewise rotations and affine maps, Nonlinearity, 14 (2001), 205-219. doi: 10.1088/0951-7715/14/2/302.  Google Scholar

[10]

J. Schmeling and S. Troubetzkoy, Interval Translation Mappings, In Dynamical systems (Luminy-Marseille, 1998), 291-302. World Sci. Publ., River Edge, NJ, 2000.  Google Scholar

[11]

H. Suzuki, S. Ito and K. Aihara, Double rotations, Discrete Contin. Dyn. Syst., 13 (2005), 515-532. doi: 10.3934/dcds.2005.13.515.  Google Scholar

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