Ergodic properties of Bogoliubov automorphisms in free probability

TL;DR

This paper investigates the ergodic properties of quantized classical dynamical systems on free Fock space, showing how classical ergodic features translate into unique ergodic, weak mixing, or mixing properties in the quantum setting.

Contribution

It establishes a detailed correspondence between classical ergodic properties and their quantum analogs in free probability, including extensions to q-commutation relations.

Findings

Quantized systems from ergodic but not weakly mixing classical systems are uniquely ergodic but not weakly mixing.

Quantized systems from weakly mixing but not mixing classical systems are uniquely weak mixing but not mixing.

Quantized systems from mixing classical systems are uniquely mixing.

Abstract

We show that some $C^*$--dynamical systems obtained by "quantizing" classical ones on the free Fock space, enjoy very strong ergodic properties. Namely, if the classical dynamical system $(X, T, \m)$ is ergodic but not weakly mixing, then the resulting quantized system $(\gg,\a)$ is uniquely ergodic (w.r.t the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system $(X, T, \m)$ which is weakly mixing but not mixing. In this case, the quantized system is uniquely weak mixing but not uniquely mixing. Finally, a quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing $C^*$--dynamical systems whose GNS representation associated to the unique invariant state generates a von Neuman factor of one of the following types: $I_{\infty}$, $II_{1}$, $III_{\l}$ where $\l\in(0,1]$. The results listed above are extended to the $q$--commutation relations, provided $|q|<\sqrt2-1$.