On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

Brandenbursky, Michael; Kędra, Jarek; Shelukhin, Egor

doi:10.1142/s0219199717500420

Cited by 18 publications

(14 citation statements)

References 19 publications

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“…Embedding Theorem. The proof of the following theorem is a variation of the proof of Ishida [19], see also [8,10]. We present it for the reader convenience.…”

Section: B2mentioning

confidence: 99%

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Entropy and quasimorphisms

Brandenbursky¹,

Marcinkowski²

2019

JMD

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Let S be a compact oriented surface. We construct homogeneous quasimorphisms on Diff(S, area), on Diff0(S, area) and on Ham(S) generalizing the constructions of Gambaudo-Ghys and Polterovich.We prove that there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area), on Diff0(S, area) and on Ham(S) whose absolute values bound from below the topological entropy. In case when S has a positive genus, the quasimorphisms we construct on Ham(S) are C 0 -continuous.We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on Ham(S) is unbounded.

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“…Embedding Theorem. The proof of the following theorem is a variation of the proof of Ishida [19], see also [8,10]. We present it for the reader convenience.…”

Section: B2mentioning

confidence: 99%

“…There exists another conjugation invariant word norm on Ham(S), the autonomous norm. It is unbounded in the case when S is a compact oriented surface, see [8,9,10,11,17]. Theorem 2 together with the fact that every autonomous diffeomorphism of a surface has entropy zero, see [29], gives a new proof of unboundedness of the autonomous norm on Ham(S).…”

mentioning

confidence: 99%

Entropy and quasimorphisms

Brandenbursky¹,

Marcinkowski²

2019

JMD

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“…REMARK 8. Another direction for comparison is quasimorphisms on surfaces that vanish on autonomous diffeomorphisms (see [4] and a series of earlier works [2,3,5]). Both approaches use quasimorphisms as tools.…”

Section: Remarkmentioning

confidence: 99%

Non-autonomous curves on surfaces

Khanevsky¹

2021

JMD

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“…Note that (cf. [9,10]) n ij (ω) is the number of times that the i-th strand overcrosses the j-th strand in the diagram of the braid β obtained by projection in the direction ω.…”

Section: Outline Of the Proofmentioning

confidence: 99%

The Lp–diameter of the group of area-preserving diffeomorphisms of S2

Brandenbursky

Shelukhin

2017

Geom. Topol.

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We show that for each p ≥ 1, the L p -metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasimorphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map X4(CP 1 ) → M0,4 ∼ = CP 1 \ {∞, 0, 1} from the configuration space of 4 points on CP 1 to the moduli space of complex rational curves with 4 marked points.

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On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

Cited by 18 publications

References 19 publications

Entropy and quasimorphisms

Entropy and quasimorphisms

Non-autonomous curves on surfaces

The Lp–diameter of the group of area-preserving diffeomorphisms of S2

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