Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

Eynard-Bontemps, Hélène; Navas, Andrés

doi:10.48550/arxiv.1912.09305

Cited by 3 publications

(12 citation statements)

References 22 publications

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“…This appears as Proposition 1.2 in [6] for the case of the interval, yet the very same proof applies to the case of the circle. Because of the equality above, asymptotic distortion is crucial in regard to the problem of approximating either the identity (in the case of the interval) or a rotation (in the case of the circle) by conjugates in the C 1+bv topology.…”

Section: Asymptotic Distortion and Conjugaciesmentioning

confidence: 64%

“…Although the Mather invariant is not a genuine circle diffeomorphism (but an equivalence class of them), the total variation of the logarithm of its derivative is well defined. The next result that relates this with the asymptotic distortion was obtained in [6] for C 2 diffeomorphisms. The proof of this extended version is given in the Appendix.…”

Section: Mather Invariant and The Fundamental Inequalitymentioning

confidence: 91%

“…If the endpoints are parabolic fixed points, then we can deform the given action into the trivial one through conjugates. This uses the ideas developed in [6] based on the notion of asymptotic distortion introduced in [14]. Indeed, triviality of the Mather invariant is equivalent to the vanishing of the asymptotic distortion, which is the key (and necessary) ingredient for implementing the conjugacy argument inside the group of C 1+ac diffeomorphisms.…”

Section: On the Technique Of Proofmentioning

confidence: 99%

“…All the other arguments are valid in C 1+ac regularity. In any case, the proof requires extending to C 1+bv diffeomorphisms the classical Szekeres construction of generating vector fields (that is, vector fields whose time-1 map is the underlying diffeomorphism), as well as some results from [6] to this context. This is carried out in the Appendix I.…”

Section: On the Technique Of Proofmentioning

confidence: 99%

“…As mentioned in §2.3, this corresponds to an extension of [6,Theorem B] to the C 1+bv setting. Assume that f (x) > x for x ∈ (0, 1) to fix ideas (otherwise, just use (2) and the fact that, by definition, the Mather invariant of f −1 equals that of f up to a reflexion.)…”

Section: Mather Invariant and The Fundamental Inequality Revisitedmentioning

confidence: 99%

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Arc-connectedness for the space of smooth $\mathbb{Z}^d$-actions on 1-dimensional manifolds

Eynard-Bontemps¹,

Navas²

2021

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We show that the space of Z d actions by C 2 orientation-preserving diffeomorphisms of a compact 1-manifold is C 1+ac arcwise connected.Centralizers of diffeomorphisms can be viewed as infinitesimal symmetries of a given dynamical system. Starting with the seminal work of Nancy Kopell [10], they have become a central object of study in dynamics. Perhaps the most relevant recent work on this is [1], which solves a longstanding question of Stephen Smale about centralizers of generic diffeomorphisms and, also, is a wonderful "window" to enter into this huge subject. However, despite the effort of many people, several natural problems remain unsolved. Here we deal with a longstanding question raised in the seventies by Harold Rosenberg [15], that partly inspired the thesis of Jean-Christophe Yoccoz [22]: Is the space of (orientation-preserving) commuting circle diffeomorphisms locally arcwise connected ? Quite surprisingly, this has revealed as a very difficult question, and only a few (and somewhat recent) results in this direction are known. These may be summarized as follows (in all what follows, all maps in consideration are assumed to preserve the orientation):Remark 1.2. The proof above actually shows more: the natural inclusion of SO(2, R) d in the space of Z d -actions on the circle (endowed with the compact-open topology) is a homotopy equivalence. (Compare [7, Propositions 4.2].) We do not know whether this result extends to higher regularity.

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Section: Asymptotic Distortion and Conjugaciesmentioning

confidence: 64%

Section: Mather Invariant and The Fundamental Inequalitymentioning

confidence: 91%

Section: On the Technique Of Proofmentioning

confidence: 99%

Section: On the Technique Of Proofmentioning

confidence: 99%

Section: Mather Invariant and The Fundamental Inequality Revisitedmentioning

confidence: 99%

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Arc-connectedness for the space of smooth $\mathbb{Z}^d$-actions on 1-dimensional manifolds

Eynard-Bontemps¹,

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Preprint

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Some examples of distorted interval diffeomorphisms of intermediate regularity

Dinamarca,

Escayola

2020

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Length functions on groups and rigidity

Ye¹

2021

Preprint

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Let G be a group. A function l : G → [0, ∞) is called a length function if (1) l(g n ) = |n|l(g) for any g ∈ G and n ∈ Z;(2) l(hgh −1 ) = l(g) for any h, g ∈ G; and(3) l(ab) ≤ l(a) + l(b) for commuting elements a, b. Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on CAT(0) spaces and Gromov δ-hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps and so on. We study length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over rings and Cremona groups. As applications, we prove that every group homomorphism from an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2, or a finite-index subgroup of the elementary group E n (R) (n ≥ 3) over an associative ring, or the Cremona group Bir(P 2 C ) to any group G having a purely positive length function must have its image finite. Here G can be outer automorphism group Out(F n ) of free groups, mapping classes group MCG(Σ g ), CAT(0) groups or Gromov hyperbolic groups, or the group Diff(Σ, ω) of diffeomorphisms of a hyperbolic closed surface preserving an area form ω.

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Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

Cited by 3 publications

References 22 publications

Arc-connectedness for the space of smooth $\mathbb{Z}^d$-actions on 1-dimensional manifolds

Arc-connectedness for the space of smooth $\mathbb{Z}^d$-actions on 1-dimensional manifolds

Some examples of distorted interval diffeomorphisms of intermediate regularity

Length functions on groups and rigidity

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