Markovian families for pseudo-Anosov flows
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914074826113024 |
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| author | Iakovoglou, Ioannis |
| author_facet | Iakovoglou, Ioannis |
| contents | Generalizing the classification approach described for transitive Anosov flows in dimension 3 in a previous preprint of the author, in this paper we describe a method for classifying (not necessarily transitive) pseudo-Anosov flows on 3-manifolds up to orbital equivalence. To every pseudo-Anosov flow $Φ$ (with no 1-prongs) on $M^3$ is associated a bifoliated plane $\mathcal{P}$ endowed with an action of $π_1(M)$. It is known that the previous action characterizes $Φ$ up to orbital equivalence and admits infinitely many Markovian families (i.e. collections of rectangles in $\mathcal{P}$ generalizing the notion of Markov partition for group actions on the plane). Our goal in this paper consists in showing that :
1) if $\mathcal{R}$ is a Markovian family of $Φ$, the number of orbits of rectangles of $\mathcal{R}$ and their pattern of intersection can be encoded by a finite combinatorial object, called a geometric type, which describes completely $Φ$ up to Dehn-Goodman-Fried surgeries on a specific finite set $Γ$ of periodic orbits of $Φ$
2) our previous choices of surgeries on $Γ$ can be read as sequences of rectangles in $\mathcal{R}$ and can be encoded by finite combinatorial objects, called cycles
3) a geometric type with cycles of $\mathcal{R}$ describes the original flow $Φ$ up to orbital equivalence
Several of the above results will be stated and proven in a slightly more general setting involving strong Markovian actions on the plane. Finally, due to the lack of bibliographic references on pseudo-Anosov flows in dimension 3, in the first part of the paper we provide an introduction to pseudo-Anosov flow theory containing several useful results for our classification approach together with their proofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Markovian families for pseudo-Anosov flows Iakovoglou, Ioannis Dynamical Systems Generalizing the classification approach described for transitive Anosov flows in dimension 3 in a previous preprint of the author, in this paper we describe a method for classifying (not necessarily transitive) pseudo-Anosov flows on 3-manifolds up to orbital equivalence. To every pseudo-Anosov flow $Φ$ (with no 1-prongs) on $M^3$ is associated a bifoliated plane $\mathcal{P}$ endowed with an action of $π_1(M)$. It is known that the previous action characterizes $Φ$ up to orbital equivalence and admits infinitely many Markovian families (i.e. collections of rectangles in $\mathcal{P}$ generalizing the notion of Markov partition for group actions on the plane). Our goal in this paper consists in showing that : 1) if $\mathcal{R}$ is a Markovian family of $Φ$, the number of orbits of rectangles of $\mathcal{R}$ and their pattern of intersection can be encoded by a finite combinatorial object, called a geometric type, which describes completely $Φ$ up to Dehn-Goodman-Fried surgeries on a specific finite set $Γ$ of periodic orbits of $Φ$ 2) our previous choices of surgeries on $Γ$ can be read as sequences of rectangles in $\mathcal{R}$ and can be encoded by finite combinatorial objects, called cycles 3) a geometric type with cycles of $\mathcal{R}$ describes the original flow $Φ$ up to orbital equivalence Several of the above results will be stated and proven in a slightly more general setting involving strong Markovian actions on the plane. Finally, due to the lack of bibliographic references on pseudo-Anosov flows in dimension 3, in the first part of the paper we provide an introduction to pseudo-Anosov flow theory containing several useful results for our classification approach together with their proofs. |
| title | Markovian families for pseudo-Anosov flows |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.19530 |