We study the conjugacy problem in the automorphism group of a regular rooted tree and in its subgroup of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in if and only if they are conjugate in , and that this problem is decidable. We prove that both conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable.
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Ievgen Bondarenko, Natalia V. Bondarenko, Said N. Sidki, Flavia R. Zapata, On the conjugacy problem for finite-state automorphisms of regular rooted trees (with an appendix by Raphaël M. Jungers). Groups Geom. Dyn. 7 (2013), no. 2, pp. 323–355