Infinite periodic points of endomorphisms over special confluent rewriting systems

Cassaigne, Julien

doi:10.5802/aif.2447

Cited by 15 publications

(13 citation statements)

References 8 publications

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“…Let φ be an eventually length-nondecreasing endomorphism of F n . In particular, it is uniformly continuous (see lemma 6.2 in [3]). Take p ∈ N such that…”

Section: The Automorphism Casementioning

confidence: 99%

“…The dynamics of free groups automorphisms is proved to be asymptotically periodic in [8]. In [3], the dynamical study of infinite fixed points was performed for monoids defined by special confluent rewriting systems (which contain free groups as a particular case). This was also achieved in [12] for virtually injective endomorphisms of virtually free groups.…”

Section: Introductionmentioning

confidence: 99%

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On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

Carvalho¹

2020

Preprint

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We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component. Considering the extension of an endomorphism to the completion we count the number of orbits for the action of the subgroup of fixed points (resp. periodic) points on the set of infinite fixed (resp. periodic) points. Finally, we study the dynamics of infinite points: for type II endomorphisms we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic. We also prove the latter for the case of automorphisms.

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“…Let φ be an eventually length-nondecreasing endomorphism of F n . In particular, it is uniformly continuous (see lemma 6.2 in [3]). Take p ∈ N such that…”

Section: The Automorphism Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

Carvalho¹

2020

Preprint

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“…holds for all i, j ≥ 0. Finally, we use (4) and (5) in similar fashion to reach the desired contradiction.…”

Section: Endomorphismsmentioning

confidence: 99%

Fixed points of endomorphisms of graph groups

Rodaro

Sykiotis

2013

Journal of Group Theory

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“…The dynamics of free groups automorphisms is proved to be asymptotically periodic in [14]. In [7], the dynamical study of infinite fixed points was performed for monoids defined by special confluent rewriting systems (which contain free groups as a particular case). This was also achieved in [19] for virtually injective endomorphisms of virtually free groups.…”

Section: Introductionmentioning

confidence: 99%

On uniformly continuous endomorphisms of hyperbolic groups

Carvalho¹

2021

Preprint

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We prove a generalization of the fellow traveller property for a certain type of quasi-geodesics and use it to present three equivalent geometric formulations of the bounded reduction property. We then provide an affirmative answer to a question from Araújo and Silva as to whether every nontrivial uniformly continuous endomorphism of a hyperbolic group with respect to a visual metric satisfies a Hölder condition. We remark that these results combined with the work done by Paulin prove that every endomorphism admitting a continuous extension to the completion has a finitely generated fixed point subgroup.

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Infinite periodic points of endomorphisms over special confluent rewriting systems

Cited by 15 publications

References 8 publications

On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion

Fixed points of endomorphisms of graph groups

On uniformly continuous endomorphisms of hyperbolic groups

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