Combinatorics of One-Dimensional Hyperbolic Attractors of Diffeomorphisms of Surfaces

A. Yu. Zhirov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 143-215.

Résumé

An algorithmic solution is given to the two following problems. Let $\Lambda _f$ and $\Lambda _g$ be one-dimensional hyperbolic attractors of diffeomorphisms $f\colon M\to M$ and $g\colon N\to N$, where $M$ and $N$ are closed surfaces, either orientable or not. Does there exist a homeomorphism $h\colon U(\Lambda _f)\to V(\Lambda _g)$ of certain neighborhoods of attractors such that $f\circ h=h\circ g$ (the topological conjugacy problem). Given $h>0$, find a representative of each class of topological conjugacy of attractors with a given structure of accessible boundary (boundary type) for which topological entropy is no greater than $h$ (the problem of enumeration of attractors). The solution of these problems is based on the combinatorial method, developed by the author, for describing hyperbolic attractors of surface diffeomorphisms.

Bibliographie

[1] Aranson S. Kh., Grines V. Z., “Topologicheskaya klassifikatsiya kaskadov na zamknutykh poverkhnostyakh”, UMN, 45:4 (1990), 3–32 | MR | Zbl

[2] Aranson S. Kh., Grines V. Z., “Kaskady na poverkhnostyakh”, Dinamicheskie sistemy – 9, Itogi nauki i tekhn. Sovr. probl. matematiki. Fund. napr., 66, VINITI, M., 1991, 148–187 | MR | Zbl

[3] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[4] Bouen R. E., Metody simvolicheskoi dinamiki, Sb. statei, Matematika. Novoe v zarubezh. nauke, 13, Mir, M., 1979

[5] Grines V. Z., “O topologicheskoi ekvivalentnosti odnomernykh bazisnykh mnozhestv diffeomorfizmov na dvumernykh mnogoobraziyakh”, UMN, 29:6 (1974), 163–164 | MR | Zbl

[6] Grines V. Z., “O topologicheskoi sopryazhennosti diffeomorfizmov dvumernogo mnogoobraziya na odnomernykh bazisnykh mnozhestvakh. Ch. 1”, Tr. Mosk. mat. o-va, 32, 1975, 35–61 | MR

[7] Grines V. Z., “O topologicheskoi sopryazhennosti diffeomorfizmov dvumernogo mnogoobraziya na odnomernykh bazisnykh mnozhestvakh. Ch. 2”, Tr. Mosk. mat. o-va, 34, 1977, 243–252 | MR | Zbl

[8] Grines V. Z., “Diffeomorfizmy na dvumernykh mnogoobraziyakh s tranzitivnymi sloeniyami”, Metody kachestvennoi teorii differentsialnykh uravnenii, no. 4, Izd. Gork. gos. un-ta, Gorkii, 1982, 62–72 | MR

[9] Grines V. Z., “Topologicheskaya klassifikatsiya strukturno ustoichivykh diffeomorfizmov s odnomernymi attraktorami i repellerami na poverkhnostyakh”, Differentsialnye uravneniya i prilozheniya, Mater. Mezhdunar. konf. (Saransk, 1994), Mordov. un-t, Saransk, 1995, 154–163

[10] Grines V. Z., “O topologicheskoi klassifikatsii strukturno ustoichivykh diffeomorfizmov poverkhnostei s odnomernymi attraktorami i repellerami”, Mat. sb., 188:4 (1997), 57–94. | MR | Zbl

[11] Grines V. Z., Zhuzhoma E. V., “O topologicheskoi klassifikatsii orientiruemykh attraktorov na $n$-mernom tore”, UMN, 34:4 (1979), 185–186 | MR

[12] Grines V. Z., Zhuzhoma E. V., “Neobkhodimye i dostatochnye usloviya topologicheskoi ekvivalentnosti orientiruemykh attraktorov na $n$-mernom tore”, Differentsialnye i integralnye uravneniya, Sb. nauch. tr., Gork. gos. un-t, Gorkii, 1981, 89–93 | MR

[13] Grines V. Z., Kalai Kh. Kh., “Diffeomorfizmy dvumernykh mnogoobrazii s prostorno raspolozhennymi bazisnymi mnozhestvami”, UMN, 40:1 (1985), 189–190 | MR | Zbl

[14] Grines V. Z., Kalai Kh. Kh., “Topologicheskaya ekvivalentnost attraktorov na dvumernykh mnogoobraziyakh”, Differentsialnye i integralnye uravneniya. Metody topologicheskoi dinamiki, Gork. gos. un-t, Gorkii, 1985, 104–105 | MR

[15] Grines V. Z., Kalai Kh. Kh., “O topologicheskoi ekvivalentnosti diffeomorfizmov s netrivialnymi bazisnymi mnozhestvami na dvumernykh mnogoobraziyakh”, Metody kachestvennoi teorii i teorii bifurkatsii, Gork. gos. un-t, Gorkii, 1988, 40–49 | MR

[16] Zhirov A. Yu., “Matritsy peresechenii odnomernykh giperbolicheskikh attraktorov na dvumernoi sfere”, Mat. zametki, 45:6 (1989), 44–55 | MR

[17] Zhirov A. Yu., “Perechislenie giperbolicheskikh attraktorov na orientiruemykh poverkhnostyakh i primeneniya k psevdoanosovskim gomeomorfizmam”, Dokl. RAN, 330:6 (1993), 683–686 | MR | Zbl

[18] Zhirov A. Yu., “Giperbolicheskie attraktory diffeomorfizmov orientiruemykh poverkhnostei. Ch. 1: Kodirovanie, klassifikatsiya i nakrytiya”, Mat. sb., 185:6 (1994), 3–50 | MR | Zbl

[19] Zhirov A. Yu., “Giperbolicheskie attraktory diffeomorfizmov orientiruemykh poverkhnostei. Ch. 2: Perechislenie i primeneniya k psevdoanosovskim diffeomorfizmam”, Mat. sb., 185:9 (1994), 29–80 | MR | Zbl

[20] Zhirov A. Yu., “Giperbolicheskie attraktory diffeomorfizmov orientiruemykh poverkhnostei. Ch. 3: Algoritm klassifikatsii”, Mat. sb., 186:2 (1995), 59–82 | MR | Zbl

[21] Zhirov A. Yu., “O minimume dilatatsii psevdoanosovskikh diffeomorfizmov krendelya”, UMN, 50:1 (1995), 197–198 | MR | Zbl

[22] Zhirov A. Yu., “Solenoidalnye predstavleniya i gomologii giperbolicheskikh attraktorov diffeomorfizmov poverkhnostei”, Mat. sb., 188:6 (1997), 3–26 | MR | Zbl

[23] Zhirov A. Yu., “Primery odnomernykh giperbolicheskikh attraktorov na neorientiruemykh poverkhnostyakh”, Mat. zametki, 65:3 (1999), 468–470 | MR

[24] Zhirov A. Yu., Plykin R. V., “Sootvetstvie mezhdu giperbolicheskimi attraktorami diffeomorfizmov poverkhnostei i obobschennymi psevdoanosovskimi diffeomorfizmami”, Mat. zametki, 58:1 (1995), 149–152 | MR | Zbl

[25] Zhuzhoma E. V., “Orientiruemye bazisnye mnozhestva korazmernosti, 1”, Izv. vuzov. Matematika, 1982, no. 5, 16–21 | MR | Zbl

[26] Kalai Kh. Kh., “Topologicheskaya ekvivalentnost diffeomorfizmov s odnomernymi nebluzhdayuschimi mnozhestvami na zamknutykh orientiruemykh poverkhnostyakh”, Differentsialnye i integralnye uravneniya. Metody topologicheskoi dinamiki, Gork. gos. un-t, Gorkii, 1985, 65–70 | MR

[27] Kalai Kh. Kh., “Topologicheskaya ekvivalentnost A-diffeomorfizmov dvumernykh mnogoobrazii na kanonicheskikh mnozhestvakh”, Metody kachestvennoi teorii i teorii bifurkatsii, Gork. gos. un-t, Gorkii, 1989, 50–61 | MR | Zbl

[28] Katok A. B., “Lokalnye svoistva giperbolicheskikh mnozhestv”, Dobavlenie 1: Nitetski Z., Vvedenie v differentsialnuyu dinamiku, Mir, M., 1975, 214–232 | MR

[29] Martin N., Inglend Dzh., Matematicheskaya teoriya entropii, Mir, M., 1988 | MR

[30] Plykin R. V., “Istochniki i stoki A-diffeomorfizmov poverkhnostei”, Mat. sb., 94:2 (1974), 243–264 | MR | Zbl

[31] Plykin R. V., “O suschestvovanii prityagivayuschikh (ottalkivayuschikh) periodicheskikh tochek A-diffeomorfizmov proektivnoi ploskosti i butylki Kleina”, UMN, 32:3 (1977), 179 | MR | Zbl

[32] Plykin R. V., “O giperbolicheskikh attraktorakh diffeomorfizmov”, UMN, 35:3 (1980), 94–104 | MR | Zbl

[33] Plykin R. V., “O giperbolicheskikh attraktorakh diffeomorfizmov (neorientiruemyi sluchai)”, UMN, 35:4 (1980), 205–206 | MR | Zbl

[34] Plykin R. V., “O geometrii giperbolicheskikh attraktorov gladkikh kaskadov”, UMN, 39:6 (1984), 75–113 | MR | Zbl

[35] Plykin R. V., “K probleme topologicheskoi klassifikatsii strannykh attraktorov dinamicheskikh sistem”, UMN, 57:6 (2002), 123–166 | MR | Zbl

[36] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989 | MR

[37] Fedotov A. G., “O solenoidakh Vilyamsa i ikh realizatsiyakh v dvumernykh dinamicheskikh sistemakh”, DAN SSSR, 252:4 (1980), 801–804 | MR | Zbl

[38] Tsishang Kh., Fogt E., Koldevai Kh.-D., Poverkhnosti i razryvnye gruppy, Nauka, M., 1988 | MR

[39] Adler R. L., Konheim A. G., McAndrew M. H., “Topological entropy”, Trans. Amer. Math. Soc., 114:2 (1965), 309–319 | DOI | MR | Zbl

[40] Arnoux P., Fathi A., “Un example de diffeomorphisme pseudo-Anosov”, C. R. Acad. Sci. Paris. Ser. 1, 312 (1991), 241–244 | MR | Zbl

[41] Arnoux P., Yoccoz J.-P., “Construction de diffeomorphismes pseudo-Anosov”, C. R. Acad. Sci. Paris. Ser. 1, 292 (1981), 75–78 | MR | Zbl

[42] Bauer M., “An upper bound for the least dilatation”, Trans. Amer. Math. Soc., 330:1 (1992), 361–370 | DOI | MR | Zbl

[43] Béguin F., Smale diffeomorphisms of surfaces: algorithm for the conjugacy problem, Preprint, Univ. Bourgogne, France, 1999

[44] Bonatti C., Langevin R., Difféomorphismes de Smale des surfaces, Astérisque, 250, Soc. Math. France, Paris, 1998 | MR | Zbl

[45] Casson A., Bleiler S., Automorphisms of surfaces after Nielsen and Thurston, LMS Student Texts, 9, Cambrige Univ. Press, Cambrige, 1988 | MR | Zbl

[46] Fathi A., “Démonstration dún theorème de Penner sur la composition des twist de Dehn”, Bull. Soc. Math. France, 120:4 (1992), 467–484 | MR | Zbl

[47] Fathi A., Laudenbach F., Poénaru V., “Travaux de Thurston sur les surfaces”, Séminaire Orsay, Astérisque, 66–67, Soc. Math. France, Paris, 1979 | MR

[48] Handel M., Thurston W. P., “New proofs of some results of Nielsen”, Adv. Math., 56:2 (1985), 173–191 | DOI | MR | Zbl

[49] Katok A., Hasselblatt B., Introduction to the modern theory of dynamical systems, Cambridge Univ. Press, Cambridge, 1995 ; Katok A. B., Khasselblat B., Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Faktorial, M., 1999 | MR

[50] Masur H., Smille J., “Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms”, Comment. Math. Helv., 68 (1993), 289–307 | DOI | MR | Zbl

[51] Penner R. C., “A construction of pseudo-Anosov homeomorphisms”, Trans. Amer. Math. Soc., 310:1 (1988), 179–197 | DOI | MR | Zbl

[52] Penner R. C., Harer J. L., Combinatorics of train tracks, Ann. Math. Stud., Princeton Univ. Press, Princeton, 1992 | MR | Zbl

[53] Rauzy G., “Echanges d'intervales el transformations induites”, Acta Arith., 34 (1979), 315–328 | MR | Zbl

[54] Robinson C., Williams R. F., “Classification of expanding attractors: an example”, Topology, 1:4 (1976), 321–323 | DOI | MR

[55] Smale S., “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 ; Smeil S., “Differentsiruemye dinamicheskie sistemy”, UMN, 25:1 (1970), 113–185 | DOI | MR | MR

[56] Williams R. F., “One-dimensional non-wandering sets”, Topology, 6:4 (1967), 473–487 | DOI | MR | Zbl

[57] Williams R. F., “The DA maps of Smale and structural stability”, Proc. Symp. Pure Math., 14 (1970), 329–334 | MR | Zbl

[58] Williams R. F., “Classification of one-dimensional attractors”, Proc. Symp. Pure Math., 14 (1970), 341–362 | MR

[59] Williams R. F., “Expanding attractors”, Publ. Math. IHES, 1973, no. 43, 161–203 | MR

[60] Zhirov A. Yu., “Complete combinatorial invariants for conjugacy of hyperbolic attractors of diffeomorphisms of surfaces”, J. Dyn. and Contr. Syst., 6:3 (2000), 397–430 | DOI | MR | Zbl