Locally Equicontinuous Dynamical Systems

A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in l∞(Z) form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems. Both of these inclusions are proper. The main result of the paper is to produce a family of examples of LE dynamical systems which are not WAP. §0. Introduction A dynamical system is a pair (X, T ) where X is a compact Hausdorff space and T a self homeomorphism. Unless stated otherwise we assume that X is metrizable and equipped with a metric d(·, ·) bounded by 1. We also assume usually that the system (X, T ) is topologically transitive and has a recurrent transitive point. The dynamical system is equicontinuous when the homeomorphisms {Tn : n ∈ Z} act on X as an equicontinuous family of maps. This class of dynamical systems is well understood. The classical theory of equicontinuous dynamical systems characterizes those systems completely. In particular we know that a topologically transitive equicontinuous system is isomorphic to a rotation of a compact monothetic group by a generator. Recently the theory of almost equicontinuous dynamical systems has been treated by several authors (see [AAB1,2],[GW]). A dynamical system (X,T ) is called almost equicontinuous (AE), if there is a point x0 ∈ X which (i) has a dense orbit, (ii) is a recurrent point and (iii) is Lyapunov stable. The latter means that x0 is an equicontinuity point (i.e. for every 2 > 0 there exists a δ > 0 such that d(x, x0) < δ implies d(Tx, Tx0) < 2, ∀n ∈ Z). It turns out that AE systems which are not equicontinuous are not at all rare. Every AE system is uniformly rigid and every uniformly rigid system has an AE cover (see definitions in the next section). However the class of AE systems in not well behaved in several ways. A subsystem as well as a factor of an AE system may fail to be AE. There is however a natural subclass of the AE systems which is well behaved. It is the class of weakly almost periodic systems (WAP) (see e.g. [EN]). Every factor as well as every subsystem of a WAP system is WAP. One way to see that the class of WAP systems is closed under these operations, as well as many others such as 1991 Mathematics Subject Classification. 54H20. Typeset by AMS-TEX 1 pointed products and inverse limits, is to see that the class of weakly almost periodic functions on Z forms a uniformly closed translation invariant subalgebra of l∞(Z). Since every WAP system is AE, the fact that the WAP property is inherited by subsystems, implies that every WAP system (X,T) has the property: • For every x ∈ X, the orbit closure Y = ŌT (x) is an AE subsystem. We take this to be the definition of a new class of dynamical systems. A dynamical system (X, T ) is called locally equicontinuous ( LE for short) if each point x ∈ X is a point of equicontinuity of the subsystem Y = ŌT (x) ⊂ X. In other words (X,T ) is LE if every transitive sub-system of X is AE. As we will show, the class of LE functions; i.e. those functions f(n) ∈ l∞(Z) that arise as the restriction of continuous functions F ∈ C(X) to the orbit of a transitive point of a LE system: