Polishability of some groups of interval and circle diffeomorphisms

Cohen, Michael P.

doi:10.4064/fm605-1-2019

Cited by 3 publications

(6 citation statements)

References 14 publications

(29 reference statements)

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“…In the article [4], it was shown that the group

{normalDiff}_{+}^{1 + A C} false(M^{1} false)

admits a unique Polish topology which is metrized by the following:

0true ρ (f, g) = sup_{x \in M^{1}} | f false(x false) - g false(x false) | + sup_{x \in M^{1}} | f^{'} (x) - g^{'} false(x false) | + \int_{M^{1}} | f^{''} - g^{''} | .

…”

Section: Maximal Pseudometrics and Quasi‐isometry Typesmentioning

confidence: 99%

“…We also wish to exhibit interesting examples of

C^{k}

‐undistorted diffeomorphisms, but we are somewhat hindered by the fact that we currently lack an explicit closed form for a maximal pseudometric on

{normalDiff}_{+}^{k} false(S^{1} false)

k ⩾ 2

. To bridge this difficulty, we employ a Polish group of ‘intermediate smoothness’, introduced in [4]: we denote by

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1 + A C} false(I false)

) the subgroup of

{normalDiff}_{+}^{1} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1} false(I false)

) consisting of diffeomorphisms whose first derivative is absolutely continuous. We have

{normalDiff}_{+}^{1} false(S^{1} false) ⩾ {normalDiff}_{+}^{1 + A C} false(S^{1} false) ⩾ {normalDiff}_{+}^{2} false(S^{1} false)

, and in [4] the author has shown that the group topology on

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

refines the

C^{1}

‐topology, but is coarser than the

C^{2}

‐topology.…”

Section: Introductionmentioning

confidence: 99%

“…To bridge this difficulty, we employ a Polish group of ‘intermediate smoothness’, introduced in [4]: we denote by

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1 + A C} false(I false)

) the subgroup of

{normalDiff}_{+}^{1} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1} false(I false)

) consisting of diffeomorphisms whose first derivative is absolutely continuous. We have

{normalDiff}_{+}^{1} false(S^{1} false) ⩾ {normalDiff}_{+}^{1 + A C} false(S^{1} false) ⩾ {normalDiff}_{+}^{2} false(S^{1} false)

, and in [4] the author has shown that the group topology on

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

refines the

C^{1}

‐topology, but is coarser than the

C^{2}

‐topology. Thus by our previous remarks, if

f \in {normalDiff}_{+}^{1 + A C} false(S^{1} false)

C^{1 + A C}

‐undistorted (that is, undistorted in the Polish group

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

), then it is also

C^{k}

‐undistorted for all

k ⩾ 2

.…”

Section: Introductionmentioning

confidence: 99%

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Maximal pseudometrics and distortion of circle diffeomorphisms

Cohen

2020

J. London Math. Soc.

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We initiate a study of distortion elements in the Polish groups normalDiff+kfalse(S1false) (1⩽k<∞), as well as normalDiff+1+ACfalse(S1false), in terms of maximal metrics on these groups. We classify distortion in the k=1 case: a C1 circle diffeomorphism is C1‐undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only nonhyperbolic fixed points which are C1+AC‐undistorted, and hence Ck‐undistorted for all k⩾2. In the Appendix, we exhibit a maximal metric on normalDiff+1+ACfalse(S1false), and observe that this group is quasi‐isometric to a hyperplane of L1false(Ifalse).

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“…In the article [4], it was shown that the group

{normalDiff}_{+}^{1 + A C} false(M^{1} false)

admits a unique Polish topology which is metrized by the following:

0true ρ (f, g) = sup_{x \in M^{1}} | f false(x false) - g false(x false) | + sup_{x \in M^{1}} | f^{'} (x) - g^{'} false(x false) | + \int_{M^{1}} | f^{''} - g^{''} | .

…”

Section: Maximal Pseudometrics and Quasi‐isometry Typesmentioning

confidence: 99%

“…We also wish to exhibit interesting examples of

C^{k}

‐undistorted diffeomorphisms, but we are somewhat hindered by the fact that we currently lack an explicit closed form for a maximal pseudometric on

{normalDiff}_{+}^{k} false(S^{1} false)

k ⩾ 2

. To bridge this difficulty, we employ a Polish group of ‘intermediate smoothness’, introduced in [4]: we denote by

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1 + A C} false(I false)

) the subgroup of

{normalDiff}_{+}^{1} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1} false(I false)

) consisting of diffeomorphisms whose first derivative is absolutely continuous. We have

{normalDiff}_{+}^{1} false(S^{1} false) ⩾ {normalDiff}_{+}^{1 + A C} false(S^{1} false) ⩾ {normalDiff}_{+}^{2} false(S^{1} false)

, and in [4] the author has shown that the group topology on

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

refines the

C^{1}

‐topology, but is coarser than the

C^{2}

‐topology.…”

Section: Introductionmentioning

confidence: 99%

“…To bridge this difficulty, we employ a Polish group of ‘intermediate smoothness’, introduced in [4]: we denote by

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1 + A C} false(I false)

) the subgroup of

{normalDiff}_{+}^{1} false(S^{1} false)

(respectively,

{normalDiff}_{+}^{1} false(I false)

) consisting of diffeomorphisms whose first derivative is absolutely continuous. We have

{normalDiff}_{+}^{1} false(S^{1} false) ⩾ {normalDiff}_{+}^{1 + A C} false(S^{1} false) ⩾ {normalDiff}_{+}^{2} false(S^{1} false)

, and in [4] the author has shown that the group topology on

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

refines the

C^{1}

‐topology, but is coarser than the

C^{2}

‐topology. Thus by our previous remarks, if

f \in {normalDiff}_{+}^{1 + A C} false(S^{1} false)

C^{1 + A C}

‐undistorted (that is, undistorted in the Polish group

{normalDiff}_{+}^{1 + A C} false(S^{1} false)

), then it is also

C^{k}

‐undistorted for all

k ⩾ 2

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Maximal pseudometrics and distortion of circle diffeomorphisms

Cohen

2020

J. London Math. Soc.

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“…Thus, our example of a C 1+AC -undistorted analytic circle diffeomorphism with no hyperbolic fixed point in Theorem 1. 4 gives a positive answer to Question 2 of [11].…”

Section: Distortion In Class C 1+acmentioning

confidence: 99%

“…In the article [4], it was shown that the group Diff 1+AC + (M 1 ) admits a unique Polish topology which is metrized by the following:…”

Section: Maximal Pseudometrics and Quasi-isometry Typesmentioning

confidence: 99%

Maximal pseudometrics and distortion of circle diffeomorphisms

Cohen

2019

Preprint

Self Cite

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We initiate a study of distortion elements in the Polish groups Diff k + (S 1 ) (1 ≤ k < ∞), as well as Diff 1+AC + (S 1 ), in terms of maximal metrics on these groups. We classify distortion in the k = 1 case: a C 1 circle diffeomorphism is C 1 -undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only non-hyperbolic fixed points which are C 1+AC -undistorted, and hence C k -undistorted for all k ≥ 2. In the appendix, we exhibit a maximal metric on Diff 1+AC + (S 1 ), and observe that this group is quasi-isometric to a hyperplane of L 1 (I).

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Generation and genericity of the group of absolutely continuous homeomorphisms of the interval

Thor¹

2022

Preprint

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We examine the Polish group H AC + of order-preserving self-homeomorphisms f of the interval [0, 1] for which both f and f −1 are absolutely continuous; in particular, we establish two results. First, we prove that H AC + is topologically 2-generated; in fact, it is generically 2-generated, i.e., there is a dense G δ set of pairs (f, g) ∈ H AC + × H AC + for which f, g is dense. Secondly, we prove that H AC + admits a dense G δ conjugacy class, and we explicitly characterize the elements thereof.

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Polishability of some groups of interval and circle diffeomorphisms

Cited by 3 publications

References 14 publications

Maximal pseudometrics and distortion of circle diffeomorphisms

Maximal pseudometrics and distortion of circle diffeomorphisms

Maximal pseudometrics and distortion of circle diffeomorphisms

Generation and genericity of the group of absolutely continuous homeomorphisms of the interval

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