Groups of Homeomorphisms of One-Manifolds, I:

Princeton University Press eBooks · 2020 · 3 citations

• Pure mathematics
• Mathematics
• Geology

Groups of Homeomorphisms of One-Manifolds, I: Actions of Nonlinear Groups

Princeton University Press eBooks · 2019-12-31 · 16 citations

This self-contained paper is part of a series [FF2, FF3] on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1. Any homomorphism of (almost any) mapping class group or automorphism group of a free group into Diff r +(S 1), r ≥ 2 is trivial. For r = 0 Nielsen showed that in many cases nontrivial (even faithful) representations exist. Somewhat weaker results are proven for finite index subgroups. 2. We construct a finitely-presented group of real-analytic diffeomorphisms of R which is not residually finite. 1

Distortion and the automorphism group of a shift

Journal of Modern Dynamics · 2018-01-01 · 8 citations

The set of automorphisms of a one-dimensional subshift $(X, σ)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy subshift. In particular, we show that the Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{{\rm top}}(X) = 0$ . We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.

The spacetime of a shift endomorphism

Transactions of the American Mathematical Society · 2017-04-20 · 2 citations

The automorphism group of a one dimensional shift space over a finite alphabet exhibits different types of behavior: for a large class with positive entropy, it contains a rich collection of subgroups, while for many shifts of zero entropy, there are strong constraints on the automorphism group. We view this from a different perspective, considering a single automorphism (and sometimes endomorphism) and studying the naturally associated two-dimensional shift system. In particular, we describe the relation between nonexpansive subspaces in this two-dimensional system and dynamical properties of an automorphism of the shift.

Rotation numbers for 𝑆² diffeomorphisms

Contemporary mathematics - American Mathematical Society · 2017-01-01 · 2 citations

These largely expository notes describe the properties of the function ${\cal R}$ which assigns a number to a $4$-tuple of distinct fixed points of an orientation preserving homeomorphism or diffeomorphism of $S^2$.

Notes on Chain Recurrence and Lyapunonv Functions

arXiv (Cornell University) · 2017-04-24 · 5 citations

This short expository note provides an introduction to the concept of chain recurrence in topological dynamics and a proof of the existence complete Lyapunov functions for homeomorphisms of compact metric spaces due to Charles Conley. I have used it as supplementary material in introductory dynamics courses.

Zero entropy subgroups of mapping class groups

Geometriae Dedicata · 2016-10-18

Rotation Numbers for $S^2$ diffeomorphisms

arXiv (Cornell University) · 2014-12-28

These largely expository notes describe the properties of the function ${\cal R}$ which assigns a number to a $4$-tuple of distinct fixed points of an orientation preserving homeomorphism or diffeomorphism of $S^2$.

Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface

Journal of Modern Dynamics · 2013-01-01 · 1 citations

We show that if $M$ is a compact oriented surface of genus $0$ and $G$ isa subgroup of Symp$^\omega_\mu(M)$ that has an infinite normal solvablesubgroup, then $G$ is virtually abelian. In particular the centralizer ofan infinite order $f \in$ Symp$^\omega_\mu(M)$ is virtually abelian. Anotherimmediate corollary is that if $G$ is a solvable subgroup ofSymp$^\omega_\mu(M)$ then $G$ is virtually abelian. We also prove aspecial case of the Tits Alternative for subgroups ofSymp$^\omega_\mu(M)$.

Triviality of some representations of MCG(S_{a}) in GL(h,ℂ),Dc``(S²) and Hig_i(𝕋²)

Proceedings of the American Mathematical Society · 2013-05-06 · 10 citations

We show the triviality of representations of the mapping class group of a genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surface in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis n comma double-struck upper C right-parenthesis comma upper D i f f left-parenthesis upper S squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>Diff</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(n,\mathbb {C}), \operatorname {Diff}(S^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m e o left-parenthesis double-struck upper T squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Homeo</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Homeo}(\mathbb {T}^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when appropriate restrictions on the genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the size of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> hold. For example, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a surface of finite type with genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g \ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper M upper C upper G left-parenthesis upper S right-parenthesis right-arrow upper G upper L left-parenthesis n comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ϕ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>MCG</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi : \operatorname {MCG}(S) \to GL(n,\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a homomorphism, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is trivial provided <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 2 g period"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n &gt; 2g.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We also show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a closed surface with genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g greater-than-or-equal-to 7"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ≥