One-parameter families of circle diffeomorphisms with strictly monotone rotation number

Kiran Parkhe

doi:10.1090/S0002-9939-2013-11699-0

One-parameter families of circle diffeomorphisms with strictly monotone rotation number

Kiran Parkhe^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We show that if f: S¹ × S¹ → S¹ × S¹ is C², with f(x, t) = (f_t(x), t), and the rotation number of f_t is equal to t for all t ∈ S¹, then f is topologically conjugate to the linear Dehn twist of the torus. We prove a differentiability result where the assumption that the rotation number of f_t is t is weakened to say that the rotation number is strictly monotone in t.

Original language	English (US)
Pages (from-to)	4327-4337
Number of pages	11
Journal	Proceedings of the American Mathematical Society
Volume	141
Issue number	12
DOIs	https://doi.org/10.1090/S0002-9939-2013-11699-0
State	Published - 2013

Keywords

Rotation number
Strict monotonicity
Topological conjugacy

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.1090/S0002-9939-2013-11699-0

Cite this

@article{1329769af4584215873ce1f9e98c9e86,

title = "One-parameter families of circle diffeomorphisms with strictly monotone rotation number",

abstract = "We show that if f: S1 × S1 → S1 × S1 is C2, with f(x, t) = (ft(x), t), and the rotation number of ft is equal to t for all t ∈ S1, then f is topologically conjugate to the linear Dehn twist of the torus. We prove a differentiability result where the assumption that the rotation number of ft is t is weakened to say that the rotation number is strictly monotone in t.",

keywords = "Rotation number, Strict monotonicity, Topological conjugacy",

author = "Kiran Parkhe",

year = "2013",

doi = "10.1090/S0002-9939-2013-11699-0",

language = "English (US)",

volume = "141",

pages = "4327--4337",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "12",

}

TY - JOUR

T1 - One-parameter families of circle diffeomorphisms with strictly monotone rotation number

AU - Parkhe, Kiran

PY - 2013

Y1 - 2013

N2 - We show that if f: S1 × S1 → S1 × S1 is C2, with f(x, t) = (ft(x), t), and the rotation number of ft is equal to t for all t ∈ S1, then f is topologically conjugate to the linear Dehn twist of the torus. We prove a differentiability result where the assumption that the rotation number of ft is t is weakened to say that the rotation number is strictly monotone in t.

AB - We show that if f: S1 × S1 → S1 × S1 is C2, with f(x, t) = (ft(x), t), and the rotation number of ft is equal to t for all t ∈ S1, then f is topologically conjugate to the linear Dehn twist of the torus. We prove a differentiability result where the assumption that the rotation number of ft is t is weakened to say that the rotation number is strictly monotone in t.

KW - Rotation number

KW - Strict monotonicity

KW - Topological conjugacy

UR - http://www.scopus.com/inward/record.url?scp=84884805215&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884805215&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2013-11699-0

DO - 10.1090/S0002-9939-2013-11699-0

M3 - Article

AN - SCOPUS:84884805215

SN - 0002-9939

VL - 141

SP - 4327

EP - 4337

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -

One-parameter families of circle diffeomorphisms with strictly monotone rotation number

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this