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$U$-system, $C$-system

A smooth dynamical system (a flow (continuous-time dynamical system) or cascade) with a compact phase manifold, which is a hyperbolic set. A diffeomorphism generating a $Y$-cascade is called a $Y$-diffeomorphism. $Y$-systems were introduced by D.V. Anosov (see , [2]), and they are often called Anosov systems.

$Y$-systems are structurally stable (cf. Rough system), and a small (in the sense of $C^1$) perturbation of a $Y$-system is again a $Y$-system. The number of periodic trajectories of a $Y$-system of period at most $T$ increases exponentially with $T$. $Y$-systems possess strong ergodic properties with respect to the wide class of so-called "Gibbs" invariant measures (see [4][6]). In particular, if a $Y$-system has a finite invariant measure "compatible with the smoothness", that is, defined in terms of local coordinates by a positive density (in the early papers, only measures of this type were considered: see –[3]), then it must be a Gibbs measure. Thus, if a $Y$-diffeomorphism does not have wandering points (cf. Wandering point), then it is metrically isomorphic to a Bernoulli automorphism: under broad assumptions the convergence of the time averages to the space average is subject to the central limit theorem, and the rate of intermixing is exponential ("exponential decay of correlations").

In the investigation of $Y$-systems, frequent use is made of symbolic dynamics, which became possible due to the Markov partitions introduced in [7], [8] (the definitive version is in [5], cf. also Symbolic dynamics). A number of results about $Y$-systems have been proved to hold also for certain other types of hyperbolic sets. There are also less direct generalizations, in which the conditions of hyperbolicity are weakened slightly (see [6], ).

Hyperbolic automorphisms of tori and geodesic flows (cf. Geodesic flow) on closed manifolds of negative curvature are $Y$-systems. There are also other examples of a related algebraic-geometric nature. In these examples, the $Y$-system has an invariant measure compatible with the smoothness. By a small perturbation, such a measure can disappear, but in view of the structural stability all points remain non-wandering. Examples of $Y$-systems with wandering points (see [9]) are of a fundamentally different character.

The existence of a $Y$-system on a manifold imposes restrictions on the topological properties of the manifold. Little is known about this in general (see [10], [11]), but the case when the stable or unstable manifold (see Hyperbolic set) is one-dimensional has been investigated thoroughly (see [9], , [13], [15]).

References

[1a] D.V. Anosov, "Roughness of geodesic flows on compact Riemannian manifolds of negative curvature" Soviet Math. Dokl. , 3 (1962) pp. 1068–1069 Dokl. Akad. Nauk SSSR , 145 : 4 (1962) pp. 707–709
[1b] D.V. Anosov, "Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature" Soviet Math. Dokl. , 4 (1963) pp. 1153–1156 Dokl. Akad. Nauk SSSR , 151 : 6 (1963) pp. 1250–1252
[2] D.V. Anosov, "Geodesic flows on closed Riemann manifolds of negative curvature" Proc. Steklov Inst. Math. , 90 (1969) Trudy Mat. Inst. Steklov. , 90 (1967)
[3] D.V. Anosov, "Some smooth ergodic systems" Russian Math. Surveys , 22 : 5 (1967) pp. 103–167 Uspekhi Mat. Nauk. , 22 : 5 (1967) pp. 107–172
[4] Ya.G. Sinai, "Gibbs measures in ergodic theory" Russian Math. Surveys , 27 : 4 (1972) pp. 21–64 Uspekhi Mat. Nauk.
[5] R. Bowen, "Equilibrium states and the ergodic theory of Anosov diffeomorphisms" , Lect. notes in math. , 470 , Springer (1975)
[6] A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" J. Soviet Math. , 7 (1977) pp. 974–1065 Itogi Nauk. i Tekhn. Mat. Anal. , 13 (1975) pp. 129–262
[7] R.L. Adler, B. Weiss, "Similarity of automorphisms of the torus" Mem. Amer. Math. Soc. , 98 (1970)
[8] Ya.G. Sinai, "Construction of Markov partitions" Funct. Anal. Appl. , 2 (1968) pp. 70–80 Funkts. Anal. i Prilozh. , 2 (1968) pp. 64–89
[9] J.M. Franks, B. Williams, "Anomalous Anosov flows" Z. Nitecki (ed.) C. Robinson (ed.) , Global theory of dynamical systems (Proc. Evanston, 1969) , Lect. notes in math. , 819 , Springer (1980) pp. 158–174
[10] M.W. Hirsch, "Anosov maps, polycyclic groups and homology" Topology , 10 : 3 (1971) pp. 177–183
[11] K. Shiraiwa, "Manifolds which do not admit Anosov diffeomorphisms" Nagoya Math. J. , 49 (1973) pp. 111–115
[12a] J.M. Franks, "Anosov diffeomorphisms" S.-S. Chern (ed.) S. Smale (ed.) , Global analysis , Proc. Symp. Pure Math. , 14 , Amer. Math. Soc. (1970) pp. 61–93
[12b] S.E. Newhouse, "On codimension one Anosov diffeomorphisms" Amer. J. Math. , 92 (1970) pp. 761–770
[12c] A. Manning, "There are no new Anosov diffeomorphisms on tori" Amer. J. Math. , 96 (1974) pp. 422–429
[13] A. Verjovsky, "Codimension one Anosov flows" Bolet. Soc. Mat. Mexicana , 19 : 2 (1974) pp. 49–77
[14a] A. Fahti, F. Laudenbach, "Les feuilletages mesurés" Astérisque , 66–67 (1979) pp. 71–126
[14b] A. Fahti, F. Laudenbach, "Comment Thurston compactifie l'espace de Teichmüller" Astérisque , 66–67 (1979) pp. 139–158
[14c] A. Fahti, V. Poénaru, "Theorème d'unicité des difféomorphismes pseudo-Anosov" Astérisque , 66–67 (1979) pp. 225–242
[15] V.V. Solodov, "Topological questions in the theory of dynamical systems" Russian Math. Surveys , 46 : 4 (1991) pp. 107–130 Uspekhi Mat. Nauk , 46 : 4 (1991) pp. 91–114

Comments

In Western literature the term Anosov system (-flow, -diffeomorphism) is consistently used. Anosov flows are particular cases of Axiom-$A$ flows, i.e., flows on compact manifolds in which the set of non-wandering points (cf. Non-wandering point) is a hyperbolic set in which the periodic points (cf. Periodic point) are dense. Axiom-$A$ flows were introduced by S. Smale. Much information about Anosov systems can be found in [a2].

References

[a1] T. Adachi, "Distribution of closed orbits with a pre-assigned homology class in a negatively curved manifold" Nagoya Math. J. , 110 (1988) pp. 1–14
[a2] R. Bowen, "On Axiom $A$ diffeomorphisms" , Amer. Math. Soc. (1978)
[a3] P. Eberlein, "When is a geodesic flow Anosov type?" J. Differential Geom. , 8 (1973) pp. 437–463
[a4] J.F. Plante, "Homology of closed orbits of Anosov flows" Proc. Amer. Math. Soc. , 37 (1973) pp. 297–300
[a5] M. Shub, "Global stability of dynamical systems" , Springer (1986)
[a6] W. De Melo, "Geometric theory of dynamical systems" , Springer (1982)
[a7] S. Smale, "Differentiable dynamical systems" Bull. Amer. Math. Soc. , 73 (1967) pp. 747–817
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Y-system. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Y-system&oldid=32282