# American Institute of Mathematical Sciences

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

## Almost every interval translation map of three intervals is finite type

 1 Kungliga Tekniska Hogskolan (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.
Citation: Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete and 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. [2] H. Bruin, Renormalization in a class of interval translation maps of $d$ branches, Dyn. Syst., 22 (2007), 11-24. doi: 10.1080/14689360601028084. [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. [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. [5] J. Buzzi, Piecewise isometries have zero topological entropy, Ergodic Theory Dynam. Systems, 21 (2001), 1371-1377. doi: 10.1017/S0143385701001651. [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. [7] A. Goetz, Sofic subshifts and piecewise isometric systems, Ergodic Theory Dynam. Systems, 19 (1999), 1485-1501. doi: 10.1017/S0143385799151964. [8] A. Goetz, Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478. [9] A. Goetz, Stability of piecewise rotations and affine maps, Nonlinearity, 14 (2001), 205-219. doi: 10.1088/0951-7715/14/2/302. [10] J. Schmeling and S. Troubetzkoy, Interval Translation Mappings, In Dynamical systems (Luminy-Marseille, 1998), 291-302. World Sci. Publ., River Edge, NJ, 2000. [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.

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##### References:
 [1] M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems, 15 (1995), 821-832. doi: 10.1017/S0143385700009652. [2] H. Bruin, Renormalization in a class of interval translation maps of $d$ branches, Dyn. Syst., 22 (2007), 11-24. doi: 10.1080/14689360601028084. [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. [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. [5] J. Buzzi, Piecewise isometries have zero topological entropy, Ergodic Theory Dynam. Systems, 21 (2001), 1371-1377. doi: 10.1017/S0143385701001651. [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. [7] A. Goetz, Sofic subshifts and piecewise isometric systems, Ergodic Theory Dynam. Systems, 19 (1999), 1485-1501. doi: 10.1017/S0143385799151964. [8] A. Goetz, Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478. [9] A. Goetz, Stability of piecewise rotations and affine maps, Nonlinearity, 14 (2001), 205-219. doi: 10.1088/0951-7715/14/2/302. [10] J. Schmeling and S. Troubetzkoy, Interval Translation Mappings, In Dynamical systems (Luminy-Marseille, 1998), 291-302. World Sci. Publ., River Edge, NJ, 2000. [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.
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