Geodesic
Interval Identification Systems and Plane Sections of 3-Periodic Surfaces
Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 72-84.

Voir la notice de l'article provenant de la source Math-Net.Ru

Interval identification systems is a notion that, on the one hand, generalizes interval exchange transformations and, on the other hand, describes special cases of such transformations. In the present paper we overview some elementary facts, address a few questions about interval identification systems, and describe explicitly systems that allow one to construct 3-periodic surfaces in the 3-space whose intersections with planes of a fixed direction have chaotic behavior. The problem of asymptotic behavior of plane sections of 3-periodic surfaces was posed by S. P. Novikov in 1982 and studied then by his students. One of the most interesting remaining open questions about such sections is reduced to the study of interval identification systems. 
@article{TRSPY_2008_263_a5,
     author = {I. A. Dynnikov},
     title = {Interval {Identification} {Systems} and {Plane} {Sections} of {3-Periodic} {Surfaces}},
     journal = {Informatics and Automation},
     pages = {72--84},
     publisher = {mathdoc},
     volume = {263},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a5/}
}
TY  - JOUR
AU  - I. A. Dynnikov
TI  - Interval Identification Systems and Plane Sections of 3-Periodic Surfaces
JO  - Informatics and Automation
PY  - 2008
SP  - 72
EP  - 84
VL  - 263
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a5/
LA  - ru
ID  - TRSPY_2008_263_a5
ER  - 
%0 Journal Article
%A I. A. Dynnikov
%T Interval Identification Systems and Plane Sections of 3-Periodic Surfaces
%J Informatics and Automation
%D 2008
%P 72-84
%V 263
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a5/
%G ru
%F TRSPY_2008_263_a5
I. A. Dynnikov. Interval Identification Systems and Plane Sections of 3-Periodic Surfaces. Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 72-84. http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a5/

[1] Keane M., “Interval exchange transformations”, Math. Ztschr., 141 (1975), 25–31 | DOI | MR | Zbl

[2] Veech W. A., “Gauss measures for transformations on the space of interval exchange maps”, Ann. Math. Ser. 2, 115:1 (1982), 201–242 | DOI | MR | Zbl

[3] Masur H., “Interval exchange transformations and measured foliations”, Ann. Math. Ser. 2, 115:1 (1982), 169–200 | DOI | MR | Zbl

[4] Hubbard J., Masur H., “Quadratic differentials and foliations”, Acta math., 142:3–4 (1979), 221–274 | DOI | MR | Zbl

[5] Agol I., Hass J., Thurston W., “The computational complexity of knot genus and spanning area”, Trans. Amer. Math. Soc., 358:9 (2006), 3821–3850 | DOI | MR | Zbl

[6] Dynnikov I., Wiest B., “On the complexity of braids”, J. Eur. Math. Soc., 9:4 (2007), 801–840 | DOI | MR | Zbl

[7] Novikov S. P., “Gamiltonov formalizm i mnogoznachnyi analog teorii Morsa”, UMN, 37:5 (1982), 3–49 | MR | Zbl

[8] Maltsev A. Ya., Novikov S.,P., “Dynamical systems, topology, and conductivity in normal metals”, J. Stat. Phys., 115 (2003), 31–46 ; arXiv: cond-mat/0312708v1[cond-mat.stat-mech] | DOI | MR

[9] Zorich A. V., “Zadacha S. P. Novikova o poluklassicheskom dvizhenii elektrona v odnorodnom magnitnom pole, blizkom k ratsionalnomu”, UMN, 39:5 (1984), 235–236 | MR | Zbl

[10] Dynnikov I. A., “Semiclassical motion of the electron. A proof of the Novikov conjecture in general position and counterexamples”, Solitons, geometry, and topology: On the crossroad, AMS Transl. Ser. 2, 179, Amer. Math. Soc., Providence, RI, 1997, 45–73 | MR | Zbl

[11] Dynnikov I. A., “Geometriya zon ustoichivosti v zadache S. P. Novikova o poluklassicheskom dvizhenii elektrona”, UMN, 54:1 (1999), 21–60 | MR | Zbl

[12] Rauzy G., “Echanges d'intervalles et transformations induites”, Acta arith., 34 (1979), 315–328 | MR | Zbl

[13] De Leo R., Dynnikov I. A., Geometry of plane sections of the infinite regular skew polyhedron $\{4,6|4\}$, E-print , 2008 arXiv: 0804.1668v1[math.GT] | MR