Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems. (English) Zbl 1045.54011

Cortez, Maria Isabel; Durand, Fabien; Host, Bernard; Maass, Alejandro

J. Lond. Math. Soc., II. Ser. 67, No. 3, 790-804 (2003).

The authors consider a topological dynamical system (TDS), that is, a couple \((X,T)\) where \(X\) is a compact metric space and \(T:X\to X\) is a homeomorphism. It is a Cantor system if \(X\) is a Cantor space, that is, \(X\) has a countable basis of its topology which consists of closed and open sets (clopen sets) and does not have isolated points. The TDS \((X,T)\) is minimal whenever \(X\) and \(\emptyset\) are the only \(T\)-invariant closed subsets of \(X\). A complex number \(c\) is said to be a continuous eigenvalue of \((X,T)\) if there exists a continuous eigenfunction associated to \(c\). If \((X,T)\) is minimal, then it is known that every continuous eigenvalue is of modulus \(1\) and every continuous function has a constant modulus. When \((X,T)\) is a TDS and \(\mu\) is a \(T\)-invariant probability measure, that is, \(T\mu=\mu\), defined on the Borel \(\sigma\)-algebra of \(X\), then the triple \((X,T,\mu)\) is called a dynamical system. A complex number \(c\) is called an eigenvalue of the dynamical system \((X,T,\mu)\) if there exists \(f \in L^2(\mu)\setminus\{0\}\) such that \(f\circ T=cf\) \(\mu\)-almost everywhere, in which case \(f\) is called an eigenfunction associated to \(c\). If the system is ergodic, then every eigenvalue is of modulus \(1\), and every eigenfunction has a constant modulus.
The authors consider only minimal Cantor systems and mainly TDSs which are uniquely ergodic, that is, systems that admit a unique invariant probability measure; this measure is then ergodic. They define the so-called linearly recurrent systems by means of a nested sequence of Kakutani-Rokhlin and prove, among other results, that these systems are uniquely ergodic but are not strongly mixing. A sufficient (necessary) condition is furnished for a complex number \(c\) to be a continuous eigenvalue (an eigenvalue, respectively) for a linearly recurrent system \((X,T,\mu)\).
A number of examples are provided where one can explicitly say that the eigenfunctions are continuous or there do not exist nontrivial eigenvalues. The paper is completed with a section where a random linearly recurrent system is studied and with some proposed questions. The motivation of this paper is that the class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts, measure-theoretical and continuous eigenvalues are the same. Then it is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems.

Reviewer: Luis Bernal Gonzales (Sevilla)

MSC:

54H20	Topological dynamics (MSC2010)
37B20	Notions of recurrence and recurrent behavior in topological dynamical systems

Keywords:

topological dynamical system; minimal Cantor system; linearly recurrent system; continuous eigenvalue; strongly mixing; weakly mixing; unique ergodicity