An obstruction to isomorphism of tensor algebras of multivariable dynamical systems

TL;DR

This paper disproves a conjecture in multivariable dynamics by providing a counterexample showing that tensor algebra isomorphism does not imply piecewise conjugacy.

Contribution

It constructs explicit counterexamples of multivariable dynamical systems with isomorphic tensor algebras that are not piecewise conjugate.

Findings

Counterexamples to the conjecture

Tensor algebra isomorphism does not imply piecewise conjugacy

Disproves the previous conjecture in multivariable dynamics

Abstract

In their paper on multivariable dynamics, Davidson and Katsoulis conjectured that two multivariable dynamical systems have isomorphic tensor algebras if and only if they are piecewise conjugate. We disprove the conjecture by constructing two piecewise conjugate multivariable dynamical systems with four maps on a two-dimensional space, whose tensor algebras are not isomorphic.