On the spectrum of dynamical systems on trees

In their paper, Schweizer and Smital (1994) [10] introduced the notions of distributional chaos for continuous maps of the interval, spectrum and weak spectrum of a dynamical system. Among other things, they have proved that in the case of continuous interval maps, both the spectrum and the weak spectrum are finite and generated by points from the basic sets. Here we generalize the mentioned results for the case of continuous maps of a finite tree. While the results are similar, the original argument is not applicable directly and needs essential modifications. In particular, it was necessary to resolve the problem of intersection of basic sets, which was a crucial point. An example of one-dimensional dynamical system with an infinite spectrum is presented. (C) 2017 Elsevier B.V. All rights reserved.