1 Introduction
We recall the concept of distortion in a group. We will say that f in a group G is a distortion element in G if it has infinite order and there exists a finite subset S of G such that:
-
(i) f belongs to the group generated by S;
-
(ii) if
$|f^n|$
is the word length of
$f^n$
in the generators of S, then
$$ \begin{align*} \lim_{n\to\infty} \frac{|f^n|}{n}=0. \end{align*} $$
This limit exists because the sequence
$(|f^n|)_{n\in \mathbb {N}}$
is sub-additive.
Remark 1.1. With this definition, we explicitly require distortion elements to be of infinite order, that is, non-torsion.
The simplest examples of groups that contain a distortion element of infinite order are the solvable Baumslag–Solitar groups:
$$ \begin{align*}BS(1; p) := \langle f, g \, : \, gfg^{-1}= f^p \rangle \quad \text{where } p \geq 2. \end{align*} $$
Indeed, by the group relation, we have for every integer
$n \geq 1$
,
$g^nfg^{-n}= f^{p^n}$
. Taking as generators the set
$S = \{f, g\}$
, we have
$ |f^{p^n}|\leq 2n+1 $
and the element f is distorted in the group
$BS(1; p)$
, with
$p \geq 2$
. Another example is the discrete Heisenberg group
$H_3$
. It is the group of integer matrices of the form
$$ \begin{align*}\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{pmatrix}.\end{align*} $$
There are generators g and h of
$H_3$
such that their commutator
$f=[g,h]$
has infinite order and generates the center of
$H_3$
. This implies that
$f^{n^2}=[g^n,h^n]$
and hence that f is a distortion element in
$H_3$
.
Before stating our main result, we recall some results about distortion elements in subgroups of the group of homeomorphisms of manifolds. Given a manifold M, we will denote by
$\operatorname {\mathrm {Diff}}^{r}_0(M)$
(respectively
$\operatorname {\mathrm {Homeo}}_0(M)$
) the connected component of the group of
$C^r$
-diffeomorphisms (respectively homeomorphisms) of M containing the identity. The following result due to Calegari and Freedman (see [Reference Calegari and Freedman5]) shows that every homeomorphism of the d-dimensional sphere is distorted.
Theorem 1.2. (Calegari and Freedman [Reference Calegari and Freedman5])
Let
$d \geq 1$
. Then, every element in the group
$\operatorname {\mathrm {Homeo}}_0(\mathbb {S}^d)$
is distorted.
In this paper, we will consider the group of diffeomorphisms of the 2-sphere. As it is cleverly noticed in [Reference Calegari and Freedman5], we have the following restrictions for a distortion element f in
$\operatorname {\mathrm {Diff}}^{1}_0(M)$
:
-
(P1) f has zero topological entropy; and
-
(P2) if x is a fixed point of f, then eigenvalues of
$Df(x)$
have absolute value
$1$
.
How can we improve property
$(P_1)$
? The authors of [Reference Calegari and Freedman5] use Oseledec’s theorem and Pesin–Ruelle inequality to prove property
$(P_1)$
. We note that recently, Navas gives a most elementary proof of property
$(P_1)$
that is valid for bi-Lipschitz homeomorphisms (see [Reference Navas14]). The following result improves the above first property
$(P_1)$
.
Theorem 1.3. (Franks and Handel [Reference Franks and Handel7])
Let M be a closed oriented surface of genus at least two. Then, there are no distortion elements in
$\operatorname {\mathrm {Diff}}^{}_0(M,\mu )$
, the subgroup of the group
$\operatorname {\mathrm {Diff}}^{1}_0(M)$
, consisting of elements that preserve a Borel probability measure on M with total support.
How can we improve property
$(P_2)$
? A priori, this property can be improved with an additional very strong hypothesis. If a distortion element f belongs to a finitely generated group that acts with a global fixed point z, then the differential at z induces a morphism of the group on
$\mathrm {GL}(2,\mathbb {R})$
and, as it is noticed in [Reference Calegari and Freedman5], the eigenvalues of
$Df(z)$
are roots of unity.
The property ‘eigenvalues are roots of unity’ is again true in some cases for our examples of groups having distortion elements without any additional hypothesis.
Case of a solvable Baumslag–Solitar group. Here, Guelman and Liousse showed the existence of a g-minimal set
$\Lambda $
contained in
$\operatorname {\mathrm {Fix}}(f)$
(see [Reference Guelman and Liousse8]). Using the fact that every point in
$\Lambda $
is recurrent, the authors of [Reference Guelman and Liousse8] proved, in [Reference Guelman and Liousse9], that for every
$z\in \Lambda $
, the eigenvalues of
$Df(z)$
are roots of unity.
Case of the Heisenberg group. If
$H_3$
acts on
$\mathbb {S}^2$
by diffeomorphisms, then Ribón ensures the existence of an orbit of cardinality at most
$2$
(see [Reference Ribón15]). Considering powers of the generating elements (if necessary), we can suppose that
$H_3$
acts with a global fixed point z. Then, Franks and Handel (see [Reference Franks and Handel7]) proved that
$Df(z)=\pm \mathrm {Id}$
.
However, we also can find infinitely many distortion elements in
$\operatorname {\mathrm {Diff}}^{1}_0(M)$
for any manifold M. More precisely, we can see the solvable Baumslag–Solitar group (defined above)
$BS(1; p)$
, with
$p \geq 2$
as a subgroup of
$\operatorname {\mathrm {Diff}}^{1}_0(M)$
.
Example 1.4. (Solvable Baumslag–Solitar group embeds in
$\operatorname {\mathrm {Diff}}^{1}_0(M)$
)
Let us consider a
$C^1$
-action of
$BS(1;p)$
on
$[0,1]$
such that the generators
$\varphi $
,
$\psi $
satisfy
$\varphi '(0)=\psi '(0)=1$
and
$\varphi '(1)=\psi '(1)=1$
(for details, see [Reference Bonatti, Monteverde, Navas and Rivas4, Lemma 4.6, p. 13], and see also [Reference Tsuboi16, paragraph following Corollary 2.3]). Then, we construct a
$C^1$
-action of
$BS(1;p)$
on
$\overline {\mathbb {D}}$
(the closed unit disk in the plane) defining (in polar coordinate)
$f_0(r,\theta ):= (\varphi (r),\theta )$
and
$g_0(r,\theta ):= (\psi (r),\theta )$
. Next, using a smooth embedding
$\phi : \overline {\mathbb {D}}\to M$
with
$ \phi (\overline {\mathbb {D}})$
being a small ball centered in some point of M, we obtain a
$C^1$
-action of
$BS(1;p)$
on M defined by
$$ \begin{align*} f(x)= \begin{cases} \phi(f_0(\phi^{-1}(x))) & \text{if } x\in \phi(\overline{\mathbb{D}}),\\ x & \text{otherwise}; \end{cases} \quad g(x)= \begin{cases} \phi(g_0(\phi^{-1}(x))) & \text{if } x\in \phi(\overline{\mathbb{D}}),\\ x & \text{otherwise}. \end{cases} \end{align*} $$
We note that this example has no recurrent point that is not fixed. Militon gives another class of distorted element in
$\operatorname {\mathrm {Diff}}^{\infty }_0(M)$
. By using Avila’s techniques (see [Reference Avila1]) and a local perfection result (see [Reference Haller, Rybicki and Teichmann10]), he obtains that every recurrent element in
$\operatorname {\mathrm {Diff}}^{\infty }_0(M)$
, that is, for which arbitrarily large iterates are arbitrarily close to the identity in the
$C^{\infty }$
-topology, is distorted.
Theorem 1.5. (Militon [Reference Militon13])
Let M be a compact manifold. Then, every recurrent element in the group
$\operatorname {\mathrm {Diff}}^{\infty }_0(M)$
is distorted.
For instance, irrational rotations of the 2-sphere are distorted (previously showed by Calegari and Freedman (see [Reference Calegari and Freedman5])). More generally, using the Anosov–Katok method (see [Reference Fathi and Herman6, Reference Herman11]), we can build recurrent elements in the case of the 2-sphere which are not conjugate to a rotation (such elements are called irrational pseudo-rotations).
We start by introducing some definitions. Let f be a homeomorphism of
$\mathbb {S}^2$
. We say that
$z\in \mathbb {S}^2$
is a (positively) recurrent point that is not fixed for f if there are arbitrarily large (positive) iterates
$f^n(z)$
that are arbitrarily close to z and
$f(z)\neq z$
. We say that a homeomorphism f of
$\mathbb {S}^2$
is an irrational (respectively rational) pseudo-rotation if:
-
(i) f has exactly two periodic points,
$z_0$
and
$z_1$
, which are both fixed; -
(ii) there exists a point z whose
$\alpha $
-limit set or
$\omega $
-limit set is not included in
$\{z_0, z_1\}$
; and -
(iii) if
$\check {f}$
is a lift of
$f|_{\mathbb {S}^2\setminus \{z_0,z_1\}}$
to the universal covering space of
$\mathbb {S}^2\setminus \{z_0,z_1\}$
, it has a unique rotation number which is irrational (respectively rational).
In this article, we exhibit some dynamical properties of distortion elements in the group of diffeomorphisms of the
$2$
-sphere. We recall that by the Lefschetz fixed point theorem, such a diffeomorphism must have at least one fixed point.
Remark 1.6. In a work in progress, the author studies the distortion elements with exactly one fixed point (the so-called Brouwer homeomorphisms). In this setting, characterizing the dynamic is more difficult, because one can construct, for instance, an infinite uncountable number of Baumslag–Solitar actions where both generators are Brouwer homeomorphisms and the oscillating set (see [Reference Béguin and Le Roux2]) of the distorted elements are all different, then such diffeomorphisms are not conjugated to each other.
Moreover, if f has a recurrent point that is not fixed, by Brouwer theorem, f must have at least two fixed points. Our main result is the following.
Theorem A. Let f be a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
that has at least two fixed points. Then, we have the following.
-
(1) If f has two fixed points and an infinite recurrent orbit, then f is an irrational pseudo-rotation and the differential at each fixed point is conjugate to an irrational rotation.
-
(2) If f has two fixed points and a distinct periodic orbit of period
$q\geq 2$
, then f is a rational pseudo-rotation. Moreover, apart from the two fixed points, all the recurrent orbits are periodic orbits with the same non-vanishing rotation number
$p/q$
. If
$q=2$
, the differential at each fixed point can be conjugate either to or to a rational rotation, and if
$$ \begin{align*} \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \end{align*} $$
$q\geq 3$
, the differential at each fixed point is conjugate to a rational rotation. Moreover, at any fixed point of
$f^q$
, the differential of
$f^q$
has only one eigenvalue which is
$1$
.
-
(3) If f has at least three fixed points, then any recurrent point of f is a fixed point and the differential of f at any fixed point has only one eigenvalue which is
$1$
.
This result is part of a bigger plan that seeks to characterize dynamics of distortion elements of the
$2$
-sphere. We will prove Theorem A using the new criterion for the existence of topological horseshoes for surface homeomorphisms that are isotopic to the identity recently found by Le Calvez and Tal (see [Reference Le Calvez and Tal12]).
2 Preliminary results
In this section, we recall some results about distortion elements in the group of diffeomorphisms of the
$2$
-sphere and homeomorphisms of the
$2$
-sphere with no topological horseshoes that we will use in the rest of the article.
2.1 Distortion elements in the group of diffeomorphisms of the
$2$
-sphere
The following result is Theorem 1.3 for the case of the
$2$
-sphere.
Theorem 2.1. [Reference Franks and Handel7, Theorem 1.3]
Let f be a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
and let
$\mu $
be an f-invariant Borel probability measure. If
$f^n$
has at least three fixed points for some smallest integer
$n\geq 1$
, then
$\operatorname {\mathrm {Supp}}(\mu ) \subset \operatorname {\mathrm {Fix}}(f^n)$
.
We state the following corollary which will be used in the rest of the article.
Corollary 2.2. Let f be a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
. Suppose that f has at least three fixed points, then:
-
(i) the set of periodic points of f coincides with that of its fixed points;
-
(ii) if X is a closed f-invariant set, then X contains a fixed point; and
-
(iii) if
$h(f)$
denotes the topological entropy of f, then
$h(f)=0$
.
2.2 Homeomorphisms of the 2-sphere with no topological horseshoes
In this section, we state some results that were proved in [Reference Le Calvez and Tal12], where the authors study some properties of homeomorphisms of the 2-sphere with no topological horseshoes. A compact subset Y of a manifold M is a topological horseshoe if it is invariant by an iterate
$f^q$
of f and if there exist a continuous map
$g:Z\to Z$
on a Hausdorff compact space Z, an integer
$N\geq 2$
, and two continuous and surjective maps (factor maps)
$p_1:Z\to Y$
and
$p_2:Z\to \{1,\ldots ,N\}^{\mathbb {Z}}$
such that:
-
(1) the factor map
$p_1$
is a semi-conjugation between g and
$f^q\vert _Y$
, where the fibers by the factor map
$p_1$
are all finite with a uniform bound in their cardinality; -
(2) the factor map
$p_2$
is a semi-conjugation between g and the Bernoulli shift
$\sigma :\{1,\ldots ,N\}^{\mathbb {Z}}\to \{1,\ldots ,N\}^{\mathbb {Z}}$
, where the preimage of every n-periodic sequence of
$\{1,\ldots ,N\}^{\mathbb {Z}}$
by the factor map
$p_2$
contains an n-periodic point of g.
Remark 2.3. Note that, if f has a topological horseshoe, then it has strictly positive topological entropy and also some power of f has periodic points. In particular, if f is a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
, then by the above property
$(P_1)$
, it has no topological horseshoes.
The first result is about rotation numbers for annular homeomorphisms with no topological horseshoes. We will write
$\mathbb {A}= (\mathbb {R}/\mathbb {Z})\times \mathbb {R}$
the open annulus, and denote by
$\pi :\mathbb {R}^2\to \mathbb {A}$
the universal covering projection of
$\mathbb {A}$
and by
$\pi _1:\mathbb {R}^2\to \mathbb {R}$
the projection in the first coordinate.
Theorem 2.4. [Reference Le Calvez and Tal12, Theorem A]
Let f be a homeomorphism of
$\mathbb {A}$
that is isotopic to the identity and let
$\check {f}$
be a lift of f to
$\mathbb {R}^2$
. We suppose that f has no topological horseshoe. Let
$\textrm {ne}^+(f)$
be the set of all point z such that its
$\omega $
-limit set (for f) is non-empty:
-
(1) each point
$z\in \textrm {ne}^+(f)$
has a well-defined rotation number
$\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
, that is, for every compact set
$ K \subset \mathbb {A}$
and every increasing sequence of integers
$(n_k)_{k\in \mathbb {N}}$
such that z and
$f^{n_k}(z)$
belong to K, we have where
$$ \begin{align*} \lim_{k\to \infty} \frac{1}{n_k}(\pi_1(\check{f}^{n_k}(\check{z})-\check{z} ) )=\operatorname{\mathrm{Rot}}_{\check{f}}(z), \end{align*} $$
$\check {z}$
is a lift of z;
-
(2) for every
$z,z'\in \textrm {ne}^+(f)$
such that
$z'$
belongs to the
$\omega $
-limit set of z, we have
$\operatorname {\mathrm {Rot}}_{\check {f}}(z')=\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
; -
(3) if
$z\in \textrm {ne}^+(f)\cap \textrm {ne}^+(f^{-1})$
is non-wandering, then
$\operatorname {\mathrm {Rot}}_{\check {f}^{-1}}(z)=-\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
; -
(4) the map
$\operatorname {\mathrm {Rot}}_{\check {f}^{\perp }}:\Omega (f)\cap (\textrm {ne}^+(f) \cup \textrm {ne}^+(f^{-1}))\to \mathbb {R}$
is continuous, where
$$ \begin{align*} \operatorname{\mathrm{Rot}}_{\check{f}^{\perp}}(z) = \begin{cases} \operatorname{\mathrm{Rot}}_{\check{f}}(z) & \text{if } z\in \Omega(f)\cap \textrm{ne}^+(f),\\ - \operatorname{\mathrm{Rot}}_{\check{f}^{-1}}(z) & \text{if } z\in \Omega(f)\cap \textrm{ne}^+(f^{-1}). \end{cases} \end{align*} $$
Theorem 2.5. [Reference Le Calvez and Tal12, Theorem F]
Let f be a homeomorphism of
$\mathbb {A}$
that is isotopic to the identity and let
$\check {f}$
be a lift of f to
$\mathbb {R}^2$
. We suppose that
$\check {f}$
is fixed point free and that there exists a positively recurrent point z such that
$\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
is well defined and equal to
$0$
. Then, f has a topological horseshoe.
Theorem 2.6. [Reference Le Calvez and Tal12, Theorem G]
Let f be an orientation-preserving homeomorphism of
$\mathbb {S}^2$
that has no topological horseshoe. Then, every non-wandering point z whose
$\alpha $
-limit set or
$\omega $
-limit set is not included in the set of fixed points of f lies in an open invariant annulus A that is fixed point free and
$f\vert _{A}$
is isotopic to the identity.
3 Main proposition
In this section, we will prove the following proposition which is key in the proof of Theorem A.
Proposition 3.1. Let f be a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
. Suppose that f has two distinct fixed points
$z_0$
and
$z_1$
, and at least another recurrent point. Let
$f_{\mathrm {ann}}$
be the homeomorphism of the closed annulus
$\overline {\mathcal {A}}$
obtained from
$\mathbb {S}^2$
by blowing up both points
$z_0$
and
$z_1$
. If
$\check {f}_{\mathrm {ann}}$
is a lift of
$f_{\mathrm {ann}}$
to the universal covering space of
$\overline {\mathcal {A}}$
, then it has a unique rotation number, that is, there exists
$\rho $
such that for each
$z\in \overline {\mathcal {A}}$
,
$$ \begin{align*} \lim_{n\to \infty} \frac{1}{n}(\pi_1(\check{f}_{\mathrm{ann}}^{n}(\check{z})-\check{z} ) )= \rho,\end{align*} $$
where
$\check {z}$
is a lift of
$z\in \overline {\mathcal {A}}$
.
To prove this proposition, we need the concept of spread introduced by Franks and Handel in [Reference Franks and Handel7].
Definition of spread. Let f be an element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
. Suppose that f has two distinct fixed points
$z_0$
and
$z_1$
. Let
$\gamma $
be a smooth curve with endpoints
$z_0$
and
$z_1$
(smooth at the endpoints), and
$\beta $
be a simple nullhomotopic closed curve on
$\mathbb {S}^2$
that crosses
$\gamma $
exactly once and such that the point
$z_1$
is contained in the disk bounded by
$\beta $
, and
$z_0$
and
$z_1$
belong to different connected components of the complement of
$\beta $
. For any curve
$\alpha $
contained in
$\mathbb {S}^2 \setminus \{z_0,z_1\}$
, the spread
$L_{\gamma ,\beta }(\alpha )$
is going to measure how many times
$\alpha $
rotates around
$\beta $
with respect to
$\gamma $
. More formally, the definition of spread is the following: let
$\overline {\mathcal {A}}$
be the compact annulus obtained from
$\mathbb {S}^2$
by blowing up both points
$z_0$
and
$z_1$
. In this case, the universal covering space of
$\overline {\mathcal {A}}$
can be identified with
$\mathbb {R}\times [0,1]$
and the covering translation can be identified by
$T(x,y)=(x+1,y)$
. For each lift
$\check {\alpha }$
of
$\alpha $
and
$\check {\gamma }$
of
$\gamma $
to
$\mathbb {R}\times [0,1]$
, there exist integers
$a<b$
such that
$\check {\alpha } \cap T^{i}(\check {\gamma })\neq \emptyset $
if and only if
$a<i<b$
. Define
$$ \begin{align*} \check{L}_{\check{\beta},\check{\gamma}}(\check{\alpha})= \max\{0, b-a-2\}\end{align*} $$
and
$$ \begin{align*} L_{\beta,\gamma}(\alpha)= \max_{\check{\alpha}}\{\check{L}_{\check{\beta},\check{\gamma}}(\check{\alpha})\}.\end{align*} $$
Define the spread of
$\alpha $
with respect to f,
$\beta $
, and
$\gamma $
as
$$ \begin{align*} \sigma_{f,\beta,\gamma}(\alpha):= \liminf_{n\to \infty} \frac{L_{\beta,\gamma}(f^n(\alpha))}{n}. \end{align*} $$
In [Reference Franks and Handel7, paragraph 5.1], the authors give some criteria for undistortion; one of them is positive spread.
Lemma 3.2. Let
$g_i$
,
$1\leq i \leq k $
, be a finite set of elements of
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
. There exists a constant
$C> 0$
such that the following property holds. If f belongs to the group generated by the
$g_i$
and if
$|f^n|$
is the word length of
$f^n$
in the generators
$g_i$
, then for all curves
$\alpha $
,
$\beta $
,
$\gamma $
, and all integers
$n>0$
,
$$ \begin{align*} L_{\beta,\gamma}(f^n(\alpha)) \leq L_{\beta,\gamma}(\alpha)+ C|f^n|. \end{align*} $$
It follows the following proposition.
Proposition 3.3. Let f be a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
. Then, for all curves
$\alpha $
,
$\beta $
,
$\gamma $
, we have
$$ \begin{align*} \sigma_{f,\beta,\gamma}(\alpha)=0.\end{align*} $$
Proof. Since f is a distortion element in
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
, there exists
$g_i$
,
$1\leq i \leq k $
, a finite set of elements of
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
such that f belongs to the group generated by the
$g_i$
. Moreover, if
$|f^n|$
is the word length of
$f^n$
in the generators
$g_i$
,
$$ \begin{align*} \lim_{n\to\infty} \frac{|f^n|}{n}=0. \end{align*} $$
According to the definition of spread and the above lemma, we have
$$ \begin{align*} \sigma_{f,\beta,\gamma}(\alpha)=\liminf_{n\to \infty} \frac{L_{\beta,\gamma}(f^n(\alpha))}{n} \leq \liminf_{n\to\infty} \frac{L_{\beta,\gamma}(\alpha)+ C|f^n|}{n}=0.\\[-41pt] \end{align*} $$
The following result is about rotation numbers for homeomorphisms of the closed annulus with no topological horseshoe.
Lemma 3.4. Let f be a homeomorphism of
$(\mathbb {R}/\mathbb {Z})\times [0,1]$
that is isotopic to the identity and let
$\check {f}$
be a lift of f to
$\mathbb {R} \times [0,1]$
. We suppose that f has no topological horseshoe. Then:
-
(1) each point
$z \in (\mathbb {R}/\mathbb {Z})\times [0,1]$
has a well-defined rotation number where
$$ \begin{align*} \operatorname{\mathrm{Rot}}_{\check{f}}(z)= \lim_{n\to \infty} \frac{\pi_1(\check{f}^n(\check{z}))- \pi_1(\check{z})}{n}, \end{align*} $$
$\check {z}$
is a lift to z;
-
(2) for every
$z,z'\in (\mathbb {R}/\mathbb {Z})\times [0,1]$
such that
$z'$
belongs to the
$\omega $
-limit set of z, we have
$\operatorname {\mathrm {Rot}}_{\check {f}}(z')=\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
; -
(3) if z is non-wandering, then
$\operatorname {\mathrm {Rot}}_{\check {f}^{-1}}(z)=-\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
; -
(4) the map
$\operatorname {\mathrm {Rot}}_{\check {f}^{\perp }}:\Omega (f)\to \mathbb {R}$
defined by
$z \mapsto \operatorname {\mathrm {Rot}}_{\check {f}}(z)$
is continuous.
Proof. Write
$f(x,y)=(f_1(x, y), f_2(x, y))$
. We can extend f to a homeomorphism
$f^{\ast }$
of the open annulus
$\mathbb {A}$
such that
$f^{\ast }(x,y)=(f_1(x,1),y)$
if
$y\geq 1$
and
$f^{\ast }(x,y)=(f_1(x,0),y)$
if
$y\leq 0$
. Like f, the homeomorphism
$f^{\ast }$
has no horseshoe. As
$ (\mathbb {R}/\mathbb {Z})\times [0,1]$
is included in a compact set of
$\mathbb {A}$
, the result follows from Theorem 2.4.
3.1 Proof of Proposition 3.1
In this section, we prove Proposition 3.1. Recall that by Remark 2.3, f, and so
$f_{\mathrm {ann}}$
, has no topological horseshoe and that by Lemma 3.4, each point
$z\in \overline {\mathcal {A}}$
has a well-defined rotation number. Suppose by contradiction that there exist z and
$z'$
(in the interior of
$\overline {\mathcal {A}}$
) such that
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z)=\rho $
and
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z')=\rho '$
, with
$\rho <\rho '$
. Conjugating if necessary, we can suppose that
$\check {f}_{\mathrm {ann}}$
is a homeomorphism of
$\mathbb {R}\times [0,1]$
and a lift of
$\gamma $
to
$\mathbb {R}\times [0,1]$
is
$\{0\}\times [0,1]$
and a lift of
$\beta $
to
$\mathbb {R}\times [0,1]$
is
$[0,1] \times \{\nu \}$
, with
$\nu>0$
small. Let
$\alpha $
be a smooth curve joining z and
$z'$
, and let
$\check {\alpha }$
be a lift of
$\alpha $
with endpoints
$\check {z}$
and
$\check {z'}$
. For
$\epsilon < (\rho '-\rho )/2$
and for every n large enough, we have
$$ \begin{align*} \pi_1(\check{f}^n_{\mathrm{ann}}(\check{z})) < n\epsilon +n\rho + \pi_1(\check{z}) \end{align*} $$
and
$$ \begin{align*} -n\epsilon +n\rho' + \pi_1(\check{z'}) < \pi_1(\check{f}^n_{\mathrm{ann}}(\check{z'})). \end{align*} $$
Therefore, we have
$$ \begin{align*} -2\epsilon n +(\rho'-\rho)n < \pi_1(\check{f}^n_{\mathrm{ann}}(\check{z'}))-\pi_1(\check{f}^n_{\mathrm{ann}}(\check{z})) \leq L_{\beta,\gamma}(f^n(\alpha))+ 2. \end{align*} $$
And so,
$$ \begin{align*} 0< \sigma_{f,\beta,\gamma}(\alpha). \end{align*} $$
This contradicts Proposition 3.3.
Now, we prove that the rotation numbers on both boundary components are equal to
$\rho $
. Suppose now that there exists z (in the interior of
$\overline {\mathcal {A}}$
) such that
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z)=\rho $
and that the rotation number of the boundary component associated to
$z_0$
is
$\rho _0$
, with
$\rho <\rho _0$
(the proof in the other case is similar). By hypothesis, we can suppose that z is recurrent and its
$\omega $
-limit set is contained in the interior of
$\overline {\mathcal {A}}$
. Suppose that every neighborhood of
$z_0$
contains a fixed point other than
$z_0$
. By the above argument, all fixed points have the same rotation number which must be
$\rho $
and, in this case,
$\rho =\rho _0$
, which contradicts our hypothesis. Therefore,
$z_0$
is an isolated fixed point. Then, we use the following result due to Birkhoff.
Lemma 3.5. (Birkhoff’s lemma, [Reference Birkhoff3, paragraph 5.1])
Let f be a homeomorphism of the plane and let
$z_0$
be an isolated fixed point for f. Then, one of the following holds.
-
(i) The point
$z_0$
has a basis of connected (closed) neighborhoods N satisfying
$f(N) \subset N $
. -
(ii) For any small enough neighborhood N of
$z_0$
, there exists a connected compact set k included in N, containing
$z_0$
and a point of
$\partial N$
, and satisfying
$f^{-1}(k)\subset k$
.
Suppose that item (i) holds. If
$z'\neq z_0$
belongs to a small connected (closed) neighborhood N satisfying
$f(N) \subset N $
, then the
$\omega $
-limit set of
$z'$
is contained in N and we have two possibilities: the
$\omega $
-limit set of
$z'$
contains
$z_0$
or the
$\omega $
-limit set of
$z'$
does not contain
$z_0$
.
Suppose that the
$\omega $
-limit set of
$z'$
contains
$z_0$
. By an argument of compactness, there exists
$z^{\prime }_0$
in the boundary component associated to
$z_0$
and in the
$\omega $
-limit set of
$z'$
. By the above argument, we have that
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z')=\rho $
and by Lemma 3.4,
$\rho =\rho _0$
, which contradicts our hypothesis.
Suppose that the
$\omega $
-limit set of
$z'$
does not contain
$z_0$
. As N is small and
$z_0$
is an isolated fixed point, we conclude that there exists a recurrent point
$z"$
that belongs to the
$\omega $
-limit set of
$z'$
and then by Lemma 3.4,
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z")=\rho $
. If this holds for all neighborhood N, we can consider a sequence
$(z^{\prime \prime }_n)_{n\in \mathbb {N}}$
of recurrent points with
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z^{\prime \prime }_n)=\rho $
for all n that converges to a point
$z^{\prime }_0$
in the boundary component associated to
$z_0$
. We use the continuity of the map rotation number (item (4) of Lemma 3.4) to deduce that
$\rho =\rho _0$
, which contradicts our hypothesis.
Suppose now that item (ii) holds. In a similar way, if
$z'\neq z_0$
belongs to a connected compact set k included in N, containing
$z_0$
and a point of
$\partial N$
, and satisfying
$f^{-1}(k)\subset k$
, then the
$\alpha $
-limit set of
$z'$
is contained in N and we again have two possibilities: the
$\alpha $
-limit set of
$z'$
contains
$z_0$
or the
$\alpha $
-limit set of
$z'$
does not contain
$z_0$
. We note first that using the recurrent point z and Lemma 3.4, we have that
$\operatorname {\mathrm {Rot}}_{\check {f}^{-1}_{\mathrm {ann}}}(z)=-\rho $
. In both cases, applying the above argument and Lemma 3.4, using
$f^{-1}$
instead of f, we obtain that
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z')=- \rho $
and
$-\rho =-\rho _0$
, which contradicts our hypothesis. This proves the result.
4 Proof of main result
In this section, we prove Theorem A. Let f be a distortion element in the group
$\operatorname {\mathrm {Diff}}^{1}_0(\mathbb {S}^2)$
that has at least two fixed points and another recurrent point. We also recall that f has no horseshoe. The proof depends on the number of fixed points of f.
Proof of Theorem A
Case 1. If f has exactly two fixed points
$z_0$
and
$z_1$
, and a positive recurrent point z whose orbit is infinite, then by Theorem 2.6, the point z lies in an open annulus A that is fixed point free, and
$f\vert _{A}$
is isotopic to the identity and so essential in the open annulus
$\mathbb {S}^2\setminus \{z_0,z_1\}$
. In this case, the positively recurrent point z in A has a well-defined rotation number
$\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
and it must be equal to
$\alpha $
(an irrational number), because otherwise, by Theorem 2.5, f must have a horseshoe which contradicts our hypothesis. Therefore, if
$\check {f}_{\mathrm {ann}}$
is a lift of
$f_{\mathrm {ann}}$
to the universal covering space of
$\overline {\mathcal {A}}$
, then by Proposition 3.1, it has a unique rotation number which must be
$\alpha $
. Therefore, the rotation number of both boundary components of
$\overline {\mathcal {A}}$
is irrational. Hence,
$Df(z_0)$
and
$Df(z_1)$
have complex eigenvalues with absolute value 1, and so the differential of f at each fixed point is conjugate to an irrational rotation.
Case 2. If f has exactly two fixed points
$z_0$
and
$z_1$
, and a distinct periodic point z of period
$q \geq 2$
, then z has a well-defined rotation number
$\operatorname {\mathrm {Rot}}_{\check {f}}(z)$
that is a rational number which we can suppose that is non-zero (see [Reference Le Calvez and Tal12, Proposition 63]). Moreover, using Corollary 2.2, we know that the set of periodic points of
$f^q$
coincides with its set of fixed points. This proves that all periodic points of f have the same period. Let
$g=f^q$
. We will prove the following classical claim.
Claim 1. Let w be a recurrent point for f. Then, w is also a recurrent point for g.
Proof. By hypothesis, there is an integer sequence
$(n_k)_{k\in \mathbb {N}}$
of positive integers that tends to
$+\infty $
and
$\lim _{k\to \infty } f^{n_k}(w)=w$
. Note that for all
$k\in \mathbb {N}$
, there exist
$m_k\in \mathbb {N}$
and
${r_k\in \{0,1,\ldots ,q-1\}}$
such that
$n_k=m_k q + r_k$
. If we consider a subsequence, we can suppose that
$r_k=r$
(r fixed) for all k, and
$$ \begin{align*} \lim\limits_{k\to \infty} f^{n_k}(w) = \lim\limits_{k\to \infty} f^{m_k q}(f^r(w)) = \lim\limits_{k\to \infty} g^{m_k}(f^r(w)) =w. \end{align*} $$
Therefore,
$w \in \omega _g(f^r(w))$
. Note that using the continuity of
$f^r$
, if
$w \in \omega _g(f^r(w))$
, then
$$ \begin{align*} f^r(w) \in \omega_g(f^{2r}(w)), f^{2r}(w) \in \omega_g(f^{3r}(w)), \ldots , f^{(q-1)r}(w) \in \omega_g(f^{qr}(w)). \end{align*} $$
Moreover, the set
$\omega _g(f^{qr}(w))$
is equal to
$\omega _g(w)$
, because it is g-invariant. Using the above inclusions and the fact that if
$x\in \omega _g(y)$
, then
$\omega _g(x) \subset \omega _g(y)$
, we obtain that
$$ \begin{align*} w \in \omega_g(f^r(w)) \subset \omega_g(f^{2r}(w)) \subset \cdots \subset \omega_g(f^{qr}(w)) = \omega_g(w). \end{align*} $$
Therefore,
$w\in \omega _g(w)$
, that is, w is a recurrent point for g.
Then, g has at least three fixed points and has no horseshoe. Moreover, using the argument below (Case 3), we conclude that all recurrent points of g lie in an open invariant annulus A that is fixed point free, and have rotation number (for g) equal to 0. From Theorem 2.5, applied to g,
$g\vert _A$
has a topological horseshoe. This contradicts Corollary 2.2 (see Remark 2.3). Therefore, w is an f periodic point of period q. Therefore, if
$\check {f}_{\mathrm {ann}}$
is a lift of
$f_{\mathrm {ann}}$
to the universal covering space of
$\overline {\mathcal {A}}$
, then by Proposition 3.1, it has a unique rotation number that must be rational non-zero. Therefore, the rotation number of both boundary components of
$\overline {\mathcal {A}}$
are rational. Hence, the eigenvalues of
$Df(z_0)$
and
$Df(z_1)$
are root of the unity, and so if
$q=2$
, the differential of f at each fixed point is either conjugate to
$$ \begin{align*} \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \quad \text{or} \quad \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \end{align*} $$
or to a rational rotation. We claim that the case where the differential of f is conjugated to
$$ \begin{align*} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \end{align*} $$
is not possible. Indeed, if it were, the map defined on the unit circle
$\mathbb {S}^1$
in the tangent plane at
$z_0$
(induced by the differential at
$z_0$
)
$u\mapsto {1}/{(\| Df(z_0) \|)} Df(z_0)\cdot u$
would have a fixed point contradicting the fact that the recurrent points of
$f_{\mathrm {ann}}$
are all periodic points of period
$2$
. Moreover, if
$q\geq 3$
, the differential at each fixed point is conjugate to a rational rotation. Moreover, at any fixed point of
$f^q$
, using the same argument, the differential of
$f^q$
has only one eigenvalue which is 1.
Case 3. If f has at least three fixed points. Suppose that z is a (positively) recurrent point of f that is not fixed. By Theorem 2.6, there exists an open annulus A that is maximal fixed point free and that contains z. As
$f\vert _A$
is isotopic to the identity, by Corollary 2.2, we can fix two fixed points
$z_0$
and
$z_1$
, one in each connected component of the complement of A. Let
$z_2$
be a third fixed point and let
$\check {f}_{\mathrm {ann}}$
be a lift of
$f\vert _{\mathbb {S}^2\setminus \{z_0,z_1\}}$
to the universal covering space of
$\mathbb {S}^2\setminus \{z_0,z_1\}$
that fixes a lift of
$z_2$
. Then, by Proposition 3.1, it has a unique rotation number that must be
$0$
. In this case, the positively recurrent point z in A has a well-defined rotation number
$\operatorname {\mathrm {Rot}}_{\check {f}_{\mathrm {ann}}}(z)$
and it must be equal to 0. From Theorem 2.5,
$f\vert _A$
has a topological horseshoe. This contradicts Corollary 2.2 (see Remark 2.3). Therefore, z is an f fixed point. Therefore, if
$\check {f}_{\mathrm {ann}}$
is a lift of
$f_{\mathrm {ann}}$
to the universal covering space of
$\overline {\mathcal {A}}$
that fixes a lift of
$z_2$
, then by Proposition 3.1, it has a unique rotation number that must be zero. Therefore, the rotation number of both boundary components of
$\overline {A}$
is zero and so there are fixed points in both boundary components of
$\overline {\mathcal {A}}$
. Hence, the only eigenvalue of
$Df(z_0)$
and
$Df(z_1)$
is 1.
This completes the proof of Theorem A.
Acknowledgements
The preparation of this article was funded by the ANID project code 11200648. I would like to thanks Patrice Le Calvez and Frédéric Le Roux for suggesting some questions which motivated Theorem A. I would also like to thank Andrés Navas for some useful comments and remarks. I am also grateful to the referee for helping to improve the main result of this article.
