# Y-system

\$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 , ), 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 ). 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 –), 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 ,  (the definitive version is in , 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 , ).

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 ) 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 , ), but the case when the stable or unstable manifold (see Hyperbolic set) is one-dimensional has been investigated thoroughly (see , , , ).

How to Cite This Entry:
Y-system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Y-system&oldid=32282