Posted by: matheuscmss | September 10, 2012

Homoclinic/heteroclinic bifurcations: thin horseshoes

Let {f:M\rightarrow M} be a {C^2}-diffeomorphism of a surface {M} with a horseshoe {K} and a periodic point {p\in K} whose stable and unstable manifolds have a quadratic tangency at some point {q\in M-K}. Consider {U} a sufficiently small neighborhood of {K} and {V} a sufficiently small neighborhood of the orbit of {q}, and let’s fix {\mathcal{U}} a sufficiently small {C^2}-neighborhood of {f} organized into the open sets {\mathcal{U}_-} and {\mathcal{U}_+} and the codimension {1} hypersurface {\mathcal{U}_0} depending on the relative positions of {W^s(p_g)} and {W^u(p_g)} near {V} (see the figure below).

Organization of the parameter space {\mathcal{U}}.

In the previous post of this series, we studied the so-called Newhouse phenomena (on the coexistence of infinitely many sinks for a residual subset of {g\in\mathcal{U}_+}) and we saw that Newhouse phenomena makes it virtually impossible to study the dynamics of all {g\in\mathcal{U}_+}.

From now on, we will be interested in the local dynamics of

\displaystyle \Lambda_g:=\bigcap\limits_{n\in\mathbb{Z}}g^{-n}(U\cup V)

for most {g\in\mathcal{U}_+}. Of course, there are plenty of reasonable ways of formalizing the notion of “most” here. For the sake of the current series of posts, we will adopt the following definition:

Definition 1 We say that a subset (i.e., a property) {\mathcal{P}\subset\mathcal{U}_+} contains (i.e., holds for) most {g\in\mathcal{U}_+} whenever for every smooth {1}-parameter family {(g_t)_{|t|<t_0}} with

  • {g_t\in\mathcal{U}_-} for {-t_0<t<0}, {g_0\in\mathcal{U}_0} and {g_t\in\mathcal{U}_+} for {0<t<t_0}, and
  • {(g_t)_{|t|<t_0}} is transverse to the codimension {1} hypersurface {\mathcal{U}_0}

one has that

\displaystyle \lim\limits_{t\rightarrow 0^+}\frac{\textrm{Leb}_1(\{s\in(0,t): g_s\in\mathcal{P}\})}{t}=1

Here, {\textrm{Leb}_1} is the {1}-dimensional Lebesgue measure.

In plain terms, {\mathcal{P}\subset \mathcal{U}_+} contains most {g\in\mathcal{U}_+} if {\mathcal{P}} has “density” {1} at {\mathcal{U}_0}, where the “density” is measured along smooth generic (i.e., transverse) {1}-parameter families crossing {\mathcal{U}_0}.

Using this reasonable notion of “most {g\in\mathcal{U}_+}”, the following question makes sense:

Is {\Lambda_g} a (uniformly hyperbolic) horseshoe for most {g\in\mathcal{U}_+}?

From our previous experience with the Newhouse phenomena, we know that this question is delicate when the horseshoe {K} is “fat”: indeed, we saw that if the Cantor sets {W^s(p)\cap K} and {W^u(p)\cap K} are thick (or, more precisely, the product of their thicknesses is strictly larger than {1}), then one has persistence of tangencies in {\mathcal{U}_+} and this is a dangerous scenario conspiring against the hyperbolicity of {\Lambda_g}. On the other hand, it is intuitive that if the horseshoe {K} is “thin” in some adequate sense, one can get rid of tangencies and this gives some hope that in this situation {\Lambda_g} is a horseshoe for most {g\in\mathcal{U}_+}.

In fact, the intuition of the previous paragraph can be formalized with the aid of the notion of Hausdorff dimension.

Definition 2 Let {A\subset\mathbb{R}^n}. Given {\mathcal{O}=\{O_i\}_{i\in I}} a countable open cover of {A}, we define its diameter {\textrm{diam}(\mathcal{O})=\sup\limits_{i\in I}\textrm{diam}(O_i)}. Given a real number {\alpha\geq 0}, the Hausdorff {\alpha}-measureof {A} is

\displaystyle m_{\alpha}(A):=\lim\limits_{\varepsilon\rightarrow 0} \,\, \inf\limits_{\substack{\mathcal{O} \textrm{ countable open cover of } A \\ \textrm{with } \textrm{diam}(\mathcal{O})<\varepsilon}} \, \, \sum\limits_{O_i\in\mathcal{O}} \textrm{diam}(O_i)^{\alpha}

and the Hausdorff dimension of {A} is {\textrm{HD}(A):=\inf\{\alpha\geq 0: m_{\alpha}(A)=0\}}.

As an exercise the reader can try to show from the definitions that the Hausdorff dimension has the following general properties:

Proposition 3 It holds:

  • (a) the Hausdorff dimension is monotone: {\textrm{HD}(B)\leq \textrm{HD}(A)} whenever {B\subset A};
  • (b) the Hausdorff dimension is countably stable: {\textrm{HD}(\bigcup\limits_{i\in\mathbb{N}} A_i)=\sup\limits_{i\in\mathbb{N}}\textrm{HD}(A_i)};
  • (c) {\textrm{HD}(A)=0} whenever {A\subset\mathbb{R}^n} is finite or countable;
  • (d) a compact {m}-dimensional submanifold {M^m\subset\mathbb{R}^n} has Hausdorff dimension {\textrm{HD}(M^m)=m};
  • (e) the Hausdorff dimension doesn’t increase under Lipschitz maps, i.e., {\textrm{HD}(f(A))\leq \textrm{HD}(A)} if {f:A\rightarrow\mathbb{R}^k} is a Lipschitz map;
  • (f) the Hausdorff dimension of a product set {A\times B} satisfies {\textrm{HD}(A\times B)\geq \textrm{HD}(A)+\textrm{HD}(B)};
  • (g) any measurable {A\subset\mathbb{R}^n} with {\textrm{HD}(A)<n} has zero Lebesgue measure.

Coming back to the study of the local dynamics of {C^2}-diffeomorphisms {g\in\mathcal{U}}, let’s consider the horseshoe {K_g}. We define the stable (resp. unstable) dimension {d_s(K_g)} (resp. {d_u(K_g)}) of {K_g} as the Hausdorff dimension of {W^u(p_g)\cap K_g} (resp. {W^s(p_g)\cap K_g}). Also, for later use, we denote by {d_s^0} and {d_u^0} the stable and unstable Hausdorff dimensions of {K=K_f}, {f\in\mathcal{U}_0}.

Remark 1 We measure the stable dimension of {K_g} using the unstable manifold of {p_g} because we’re interested in the transverse structure of the stable set of {K_g}. Also, we call {d_s(K_g)} the stable dimension of {K_g} instead of stable dimension of {K_g} at {p_g} because it is possible to prove that {d_s(K_g)=W^u(x)\cap K_g} for all {x\in K_g}

The stable and unstable dimensions {d_s(K_g)} and {d_u(K_g)} are nicely related to the geometry of the horseshoe {K_g} because of the formula:

\displaystyle \textrm{HD}(K_g)=d_s(K_g)+d_u(K_g)

Remark 2 The idea behind this formula is that at small scales the horseshoe {K_g} looks like the product of the regular Cantor sets {W^s_{loc}(x)\cap K_g} and {W^u_{loc}(x)\cap K_g} and this allows to get an improved version of item (f) of Proposition 3 in this case.

Using the notion of stable and unstable dimensions, S. Newhouse, J. Palis and F. Takens (see here and here) in 1987 proved the following theorem:

Theorem 4 (S. Newhouse, J. Palis and F. Takens) Suppose that {d_s^0+d_u^0<1} for {f\in\mathcal{U}_0}. Then, {\Lambda_g} is a uniformly hyperbolic horseshoe for most {g\in\mathcal{U}_+}.

Informally speaking, this theorem says that if the horseshoe {K=K_f} is thin enough in the sense that {\textrm{HD}(K)=d_s^0+d_u^0<1}, then {\Lambda_g} is a horseshoe for most ways of unfolding the quadratic tangency of {f} at {q} (i.e., for most {g\in\mathcal{U}_+}).

Let us now explain why this theorem is intuitively plausible. Let us consider {(g_t)_{|t|<t_0}} a smooth {1}-parameter family transverse to {\mathcal{U}_0} at {f=g_0}. Of course, our first obstacle towards hyperbolicity is the issue of tangencies. So, using the notations from Section 1 of the previous post, let’s again consider the regular Cantor sets {K^s(g_t)} and {K^u(g_t)} on the line of tangencies {\ell(g_t)} whose intersections {K^s(g_t)\cap K^u(g_t)} accounts for all tangencies between the stable and unstable laminations of {K_{g_t}}. Because the tangency for {f\in\mathcal{U}_0} are quadratic, by adequate reparametrization, we may think that the regular Cantor sets {K^s(g_t)} and {K^u(g_t)} live in the real line {\mathbb{R}} and they move with unit speed relatively to each other, i.e.,

\displaystyle K^s(g_t)=K^s(g_0) \quad \textrm{ and } \quad K^u(g_t)=K^u(g_0)+t

for all {|t|<t_0}. In this context, note that {K^s(g_t)\cap K^u(g_t)\neq\emptyset} if and only if {t\in K^s(g_0)\ominus K^u(g_0)} where {A\ominus B:=\{x-y: x\in A,\, y\in B\}} denotes the arithmetic difference between {A\subset\mathbb{R}} and {B\subset\mathbb{R}}. In other words, the arithmetic difference {K^s(g_0)\ominus K^u(g_0)} of the regular Cantor sets {K^s(g_0)} and {K^u(g_0)} accounts for all parameters {t\in(-t_0,t_0)} such that the stable and unstable laminations of {K_{g_t}} exhibits some tangency. Therefore, it is desirable to know the size of this arithmetic difference.

In this direction, one observes that {K^s(g_0)\ominus K^u(g_0)=\pi(K^s(g_0)\times K^u(g_0))} where {\pi:\mathbb{R}^2\rightarrow\mathbb{R}} is the projection {\pi(x,y)=x-y}. Since {K^s(g_0)} and {K^u(g_0)} are regular Cantor sets of Hausdorff dimensions {d_s^0} and {d_u^0}, one has that the product set {K^s(g_0)\times K^u(g_0)} has Hausdorff dimension {d_s^0+d_u^0}.

Remark 3 This holds because {W^s_{loc}(p)\cap K} and {W^u_{loc}(p)\cap K} are Cantor sets of Hausdorff dimensions {d_s^0} and {d_u^0} by definition, and {K^s(g_0)} and {K^u(g_0)} are diffeomorphic to {W^s_{loc}(p)\cap K} and {W^u_{loc}(p)\cap K}, so that item (e) of Proposition 3 says that {K^s(g_0)} and {K^u(g_0)} have Hausdorff dimensions {d_s^0} and {d_u^0}. Furthermore, since {W^s_{loc}(p)\cap K} and {W^u_{loc}(p)\cap K} are regular Cantor sets, it is possible to show that {K^s(g_0)\times K^u(g_0)} has Hausdorff dimension {d_s^0+d_u^0}.

By item (e) of Proposition 3, we obtain that the arithmetic difference {K^s(g_0)\ominus K^u(g_0)=\pi(K^s(g_0)\times K^u(g_0))} has Hausdorff dimension

\displaystyle \textrm{HD}(K^s(g_0)\ominus K^u(g_0))\leq d_s^0+d_u^0(<1)

because the projection {\pi} is Lipschitz. By item (g) of Proposition 3, we conclude that the arithmetic difference {K^s(g_0)\ominus K^u(g_0)\subset\mathbb{R}} has zero Lebesgue measure. In other words, the assumption {d_s^0+d_u^0<1} imposes a severe restriction on the set of parameters {|t|<t_0} such that the invariant laminations of {K_{g_t}} exhibits a tangency, namely, these parameters have zero Lebesgue measure.

At this point, we got rid of the issue of tangencies (from the measure-theoretical point of view), but unfortunately this per se is not sufficient to ensure the hyperbolicity of {\Lambda_g}: indeed, while it is quite clear that the pieces of orbits passing near the horseshoe {K_g} have natural canditates for the stable and unstable directions {E^s(x)} and {E^u(x)} in Definition 1 (of hyperbolicity) in this previous post, this is not so clear in the region {V} (near the quadratic tangency for {f\in\mathcal{U}_0}) as the candidate directions {E^s(x)} and {E^u(x)} may reverse their role (and thus the hyperbolicity is lost) due to almost tangencies between the invariant laminations of {K_g}. In other words, we need not only to ensure that {K^s(g_t)\cap K^u(g_t)=\emptyset}, but we also have to ensure that {K^s(g_t)} and {K^u(g_t)} are sufficiently far apart from each other in order to obtain the hyperbolicity of {\Lambda_{g_t}}.

To formalize the idea of the previous paragraph, one needs to know the localization of points of {\Lambda_{g_t}-K_{g_t}}, that is, the points of {\Lambda_{g_t}} whose {g_t}-orbit passes by the region {V}. Here, one can show (see the proposition by the end of page 213 of Palis-Takens book) that, given {c>0}, all points of {\Lambda_{g_t}-K_{g_t}} have some {g_t}-iterate in {V} at a distance {\leq c\cdot t} of the invariant laminations of {K_{g_t}} for all {t} sufficiently small (depending on {c>0}). This fact is very interesting because it says that one can understand the orbits in {\Lambda_{g_t}-K_{g_t}} by looking at the intersection of the {c\cdot t}-neighborhoods {\mathcal{F}^s(t)} and {\mathcal{F}^u(t)} of the stable and unstable laminations of {K_{g_t}}. We illustrate this intersection in the figure below:

Neighborhoods {\mathcal{F}^s(t)}/{\mathcal{F}^u(t)} indicated by horizontal/parabolic-like strips. The intersection {\mathcal{F}^s(t)\cap\mathcal{F}^u(t)} is indicated by strong grey.

In this picture, we are again using the fact that the tangencies for {f\in\mathcal{U}_0} are quadratic. From this figure, we see that the geometry of {\mathcal{F}^s(t)\cap\mathcal{F}^u(t)} is controlled by the relative position of the {c\cdot t}-neighborhoods {A_{c\cdot t}} and {B_{c\cdot t}} of the Cantor sets {K^s(g_t)} and {K^u(g_t)} on the line of tangency {\ell(g_t)}: for instance, if the distance between {A_{c\cdot t}} and {B_{c\cdot t}} is {\geq 2c\cdot t}, then the angle between the leaves of the stable and unstable foliations at any point {x\in \mathcal{F}^s(t)\cap\mathcal{F}^u(t)} is {\geq \sqrt{2c\cdot t}}. See the figure below.

Angle estimate in the region {\mathcal{F}^s(t)\cap\mathcal{F}^u(t)} (here {s:=2c\cdot t}).

In other terms, if the distance between the sets {A_{c\cdot t}} and {B_{c\cdot t}} is {\geq 2c\cdot t}, then we don’t see almost tangencies, i.e., the angle between leaves of the stable and unstable foliations is {\geq \sqrt{2ct}}. Since the tangents to the leaves of the stable and unstable foliations at {\Lambda_g-K_g} are the natural candidates for stable and unstable directions over {\Lambda_{g_t}} in the sense of Definition 1 in this previous post, it is not surprising that one can actually prove that {\Lambda_{g_t}} is a hyperbolic set (and hence a horseshoe) whenever this angle estimate holds. More precisely, one has the following proposition:

Proposition 5 Suppose that the distance between the sets {A_{c\cdot t}} and {B_{c\cdot t}} is {\geq 2c\cdot t}. Then, {\Lambda_{g_t}} is a hyperbolic set (and hence a horseshoe).

We will come back to this Proposition 5 after we end the argument showing Theorem 4.

In resume, we have that if the distance {d(A_{c\cdot t},B_{c\cdot t})} between {A_{c\cdot t}} and {B_{c\cdot t}} is {\geq 2c\cdot t}, then {\Lambda_{g_t}} is a horseshoe. Thus, it remains only to estimate the Lebesgue measure of the set of parameters {t} such that {d(A_{c\cdot t},B_{c\cdot t})<2c\cdot t} in order to deduce that {\Lambda_{g}} is a horseshoe for most {g\in\mathcal{U}}. At this stage, one notes that {A_{c\cdot t}} is the {c\cdot t}-neighborhood of {K^s(g_t)=K^s(g_0)} and {B_{c\cdot t}} is the {c\cdot t}-neighborhood of {K^u(g_t)=K^s(g_0)+t}, and one recalls that, by Proposition 3 at page 104 of Palis-Takens book (whose proof occupies slightly less than half of page 105 of this book), the fact that the regular Cantor sets {K^s(g_0)} and {K^u(g_0)} satisfy {d_s^0+d_u^0<1} implies that for all {\varepsilon>0} there exists {c=c(\varepsilon)>0} and {t(\varepsilon)>0} such that

\displaystyle \frac{\textrm{Leb}_1(\{s\in(0,t): d(A_{c\cdot s},B_{c\cdot s})<2c\cdot s\})}{t}<\varepsilon

for each {0<t<t(\varepsilon)}. Of course, the reader has no difficulty to recognize that this last estimate readily implies that {\Lambda_g} is horseshoe for most {g\in\mathcal{U}_0}, and thus the sketch of proof of Theorem 4 is complete modulo Proposition 5.

At this stage, our understanding of homoclinic bifurcations of quadratic tangencies associated to thin horseshoes is considerably advanced. Hence, the next natural step is to study the bifurcations associated to fat horseshoes, and, as it turns out, this will be the theme of the next posts. For now, let us close today’s post by giving a sketch of proof of Proposition 5.

We start the argument by recalling the so-called invariant cone field criterion for the hyperbolicity of compact invariant sets. Roughly speaking, this criterion says that a compact {g}-invariant set {\Lambda} is hyperbolic whenever we can find approximately invariant directions that are uniformly contracted and expanded.

More concretely, suppose that we can guess where the stable and unstable directions {E^s} and {E^u} are located in the following sense: we have a continuous family of directions {E(x), F(x)\subset T_xM} for each {x\in \Lambda} that are complementary, i.e., {E(x)\oplus F(x)=T_xM} such that, for some positive continuous function {a:\Lambda\rightarrow\mathbb{R}^+} and some constants {C>0}, {0<\theta<1}, {\rho>1}, the continuous families of cones

\displaystyle \mathcal{C}^s_a(x):=\{v\in T_xM: v=e+f, e\in E(x), f\in F(x), \|f\|\leq a(x)\|e\| \},

\displaystyle \mathcal{C}^u_a(x):=\{v\in T_xM: v=e+f, e\in E(x), f\in F(x), \|e\|\leq a(x)\|f\| \}

for each {x\in\Lambda} satisfy

  • (CFa) {Dg(x)(\mathcal{C}^u_a(x))\subset \mathcal{C}^u_{\theta\cdot a}(f(x))}, {Dg^{-1}(x)(\mathcal{C}^s_a(x))\subset \mathcal{C}^s_{\theta\cdot a}(g^{-1}(x))}, and
  • (CFb) {\|Dg^n(x)\cdot v\|\geq C\rho^n\|v\|} and {\|Dg^{-n}(x)\cdot w\|\geq C\rho^n\|w\|} for every {v\in\mathcal{C}^u_a(x)}, {w\in \mathcal{C}^s_a(x)} and {n\in\mathbb{N}}.

Then, the invariant cone field criterion says that {\Lambda} is a hyperbolic set. Intuitively, the conditions from the invariant cone field criterion imply that

\displaystyle E^s(x):=\bigcap\limits_{n\in\mathbb{N}} Dg^{-n}(g^n(x))(\mathcal{C}^s_a(g^n(x))) \textrm{ and } E^u(x):=\bigcap\limits_{n\in\mathbb{N}} Dg^n(g^{-n}(x))(\mathcal{C}^u_a(g^{-n}(x)))

are the stable and unstable directions because the “widths” of the nested sequence of cones {Dg^n(g^{-n}(x))(\mathcal{C}^u_a(g^{-n}(x)))\subset T_xM} decrease (exponentially) in view of item (CFa) and this can be used to prove that this sequence converges to a subspace {E^u(x)} whose vectors are expanded because of item (b). See, e.g., Hasselblat-Katok book for more discussion on the invariant cone field criterion (and for a proof of this result).

In our context, we wish to show the hyperbolicity of {\Lambda_{g_t}}, and, using our previous notations, it suffices to construct stable and unstable directions for the orbits in {\Lambda_{g_t}-K_{g_t}}. By the invariant cone field criterion, our task is reduced to guess where such directions are located, that is, to exhibit adequate cones along the orbits in {\Lambda_{g_t}-K_{g_t}}. In this direction, we recall that every orbit in {\Lambda_{g_t}-K_{g_t}} passes by {\mathcal{F}^s(t)\cap\mathcal{F}^u(t)} (i.e., {\Lambda_{g_t}-K_{g_t}\cap V\subset \mathcal{F}^s(t)\cap\mathcal{F}^u(t)}, so that our goal is to study the “tangent bundle dynamics” of {g} along orbits of points {z\in \mathcal{F}^s(t)\cap\mathcal{F}^u(t)\cap (\Lambda_{g_t}-K_{g_t})}.

For the sake of concreteness, we will assume that the dynamics of {g=g_t} is “linear” in the following sense. We assume that {M=\mathbb{R}^2} (i.e., we are studying {M} via an adequate system of coordinates), {p=(0,0)\in\mathbb{R}^2} is a hyperbolic fixed point of {g}, {V\subset\mathbb{R}^2} is a small neighborhood of the point {q=(1,0)} (of quadratic tangency of {f\in\mathcal{U}_0}) and, given any {z\in V}, {g} and its first {m} iterates act linearly as the matrix {A}, {A(x,y):=(\lambda x, \sigma y)} where {0<\lambda<1<\sigma} are the eigenvalues of {A:=Dg(p)}, until {g^m(z)} hits a small neighborhood {W} of {r=(1,0)=g^{-1}(q)}. Then, if {z\in(\Lambda_{g_t}-K_{g_t})}, {g} sends {g^m(z)} to a point {g^{m+1}(z)\in \mathcal{F}^s(t)\cap\mathcal{F}^u(t)\cap (\Lambda_{g_t}-K_{g_t})} and {B=Dg(g^m(z))} is a matrix sending a direction of the form {(x^s,1)}, {2\sqrt{t}\leq |x^s|\leq 1} into the horizontal direction {(1,0)}, and the vertical direction {(0,1)} into a direction of the form {(1, y^u)} where {2\sqrt{t}\leq |y^u|\leq 1}. It is not hard to convince oneself of “naturality” of these assumptions by staring at the following picture while taking some coffee:

Action of {g} after simplifications: {g} acts linearly as a hyperbolic matrix {A} except for the transition from {g^m(z)} to {g^{m+1}(z)}, where (the derivative of) {g} acts a a matrix sending the red/blue direction at {g^m(z)} into the red/blue direction at {g^{m+1}(z)}. Note that, as indicated in the figure, the angle between red and blue directions is {\sim\sqrt{t}}.

In this picture, we are strongly using the main assumption of Proposition 5 (on the distance between the {A_{ct}} and {B_{ct}}).

By symmetry, it suffices to construct unstable cone fields {\mathcal{C}^u(z)} for {z\in \mathcal{F}^s(t)\cap\mathcal{F}^u(t)\cap (\Lambda_{g_t}-K_{g_t})}. We consider the directions {E=\mathbb{R}\cdot (1,0)} and {F=\mathbb{R}\cdot v^u} where {v^u:=(1,y^u)}, and we define

\displaystyle \mathcal{C}^u_{1/2}(z):=\{v=a\cdot v^u + b\cdot (1,0): |b|\leq (1/2)|a|\}

Observe that {z\in V} implies that {z} is very close to {(1,0)}. Hence, denoting by {m=m(z)} the number of {g}-iterates such that {g^m(z)\in W}, we have that {m\geq\log(1/t)/\log(\sigma)}. Indeed, since {g} acts linearly as {g(x,y)=(\lambda x,\sigma y)} until we hit the small neighborhood {W} of {(0,1)}, denoting by {z=(x,y)}, we have that {(\lambda^m x, \sigma^m y)=g^m(z)\in W} essentially implies that {\sigma^m y=1}. Since {z\in \mathcal{F}^s(t)\cap\mathcal{F}^u(t)\cap (\Lambda_{g_t}-K_{g_t})} implies that {|y|\leq t}, we conclude from {\sigma^m y=1} that {m\geq \log(1/t)/\log(\sigma)}.

During the period {1\leq n\leq m}, the derivative {A} of {g} evolves the cone {\mathcal{C}^u_{1/2}(z)} in a very simple way:

\displaystyle A^n(\mathcal{C}^u_{1/2}(z))\subset \{(\hat{x},\hat{y})\in\mathbb{R}^2: |\hat{x}|\leq (3/4)\cdot(\lambda/\sigma)^n \cdot (1/\sqrt{t})\cdot |\hat{y}|\}

because, given {v=a\cdot v^u + b\cdot (1,0) \in\mathcal{C}^u_{1/2}(z)} (i.e., {|b|\leq |a|/2}), {A^n(v)=A^n(a+b, ay^u)=(\lambda^n(a+b), \sigma^n a y^u)}, so that

\displaystyle \begin{array}{rcl} |\lambda^n(a+b)|&\leq& \lambda^n(|a|+|b|)\leq (3/2)\lambda^n|a| = (3/4)(\lambda/\sigma)^n\frac{2}{|y^u|}|\sigma^n a y^u| \\ &\leq& (3/4)\cdot (\lambda/\sigma)^n\cdot (1/\sqrt{t})\cdot |\sigma^n a y^u| \end{array}

i.e., {A^n(v)\in\{(\hat{x},\hat{y})\in\mathbb{R}^2: |\hat{x}|\leq (3/4)\cdot(\lambda/\sigma)^n \cdot (1/\sqrt{t})\cdot |\hat{y}|\}} whenever {v\in\mathcal{C}^u_{1/2}(z)}. Here, we used in the last inequality the fact that {|y^u|\geq 2\sqrt{t}}.

Therefore, by defining {\mathcal{C}^u_{\varepsilon}(g^n(z)):=\{(\hat{x},\hat{y})\in\mathbb{R}^2: |\hat{x}|\leq \varepsilon\cdot |\hat{y}|\}} for {z\in V} and {1\leq n\leq m=m(z)}, we have that {Dg(g^{n-1}(z))(\mathcal{C}^u_{(\lambda/\sigma)^{n-1}(1/\sqrt{t})}(g^{n-1}(z)))\subset \mathcal{C}^u_{(3/4)(\lambda/\sigma)^n(1/\sqrt{t})}(g^n(z))}. In particular, the item (CFa) from the invariant cone field criterion is automatically satisfied (with {a(g^n(z))=(\lambda/\sigma)^n(1/\sqrt{t})} and {\theta=3/4}) during the first {m} iterates of {g}! In other words, as far as item (CFa) is concerned, we have nothing to check for the first {m} iterates of {g}!

On the other hand, at time {m+1}, it is no longer true that item (CFa) is automatic: indeed, since {g^{m+1}(z)\in V}, the point {g^{m+1}(z)} was already assigned a cone {\mathcal{C}^u_{1/2}(g^{m+1}(z))}, and, thus, the item (CFa) asks us to show that {Dg(g^m(z)):=B} sends the cone {\mathcal{C}^u_{(\lambda/\sigma)^m(1/\sqrt{t})}(g^m(z))} inside {\mathcal{C}^u_{\theta/2}(g^{m+1}(z))} for some constant {0<\theta<1}.

In this direction, let us recall that {m\geq \log(1/t)/\log(\sigma)}. Thus, {1/\sigma^m\leq t} and, a fortiori,

\displaystyle (\lambda/\sigma)^m(1/\sqrt{t})\leq t^{\log(1/\lambda)/\log\sigma}\sqrt{t}.

Therefore, any {v=(\hat{x},\hat{y})\in\mathcal{C}^u_{(\lambda/\sigma)^m(1/\sqrt{t})}(g^m(z))} satisfies

\displaystyle |\hat{x}|\leq (\lambda/\sigma)^m(1/\sqrt{t})|\hat{y}|\leq t^{\log(1/\lambda)/\log\sigma}\sqrt{t}|\hat{y}| \ \ \ \ \ (1)

Next, we recall that {B(x^s,1)=(1,0)} and {B(0,1)=(1,y^u)=v^u}, where {2\sqrt{t}\leq |x^s|\leq 1} and {2\sqrt{t}\leq |y^u|\leq 1}. In particular, given {v=(\hat{x}, \hat{y})\in\mathbb{R}^2}, we have

\displaystyle \begin{array}{rcl} B(v)&=&\hat{x}\cdot B(1,0)+\hat{y}\cdot v^u = \frac{\hat{x}}{x^s}\cdot B((x^s,1)-(0,1))+\hat{y}\cdot v^u \\ &=& \frac{\hat{x}}{x^s}\cdot((1,0)-v^u)+\hat{y}\cdot v^u \\ &=& a\cdot v^u + b\cdot (1,0) \end{array}

where {a=\hat{y}-\hat{x}/x^s} and {b=\hat{x}/x^s}.

It follows that, if {v=(\hat{x},\hat{y})\in\mathcal{C}^u_{(\lambda/\sigma)^m(1/\sqrt{t})}(g^m(z))}, then {B(v)=a\cdot v^u+ b\cdot (1,0)}, where

\displaystyle \begin{array}{rcl} |b|&=&|\hat{x}|/|x^s|\leq t^{\log(1/\lambda)/\log\sigma}\sqrt{t}|\hat{y}|/(2\sqrt{t}) = t^{\log(1/\lambda)/\log\sigma}|\hat{y}|/2 \\ &\leq & t^{\log(1/\lambda)/\log\sigma}|\hat{y}-\hat{x}/x^s|= t^{\log(1/\lambda)/\log\sigma}|a| \end{array}

Here, we used (1) and {|x^s|\geq 2\sqrt{t}} to deduce that {|\hat{x}|/|x^s|\leq t^{\log(1/\lambda)/\log\sigma}|\hat{y}|/2\leq |\hat{y}/2|} for {0<t<1} and, a fortiori, {|\hat{y}|/2\leq |\hat{y}-\hat{x}/x^s|}.

In resume, we showed that, if {v=(\hat{x},\hat{y})\in\mathcal{C}^u_{(\lambda/\sigma)^m(1/\sqrt{t})}(g^m(z))}, then {B(v)=a\cdot v^u+ b\cdot (1,0)}, where

\displaystyle |b|\leq t^{\log(1/\lambda)/\log\sigma}|a|.

In particular, {B(v)\in\mathcal{C}^u_{3/8}(g^{m+1}(z))} (={\mathcal{C}^u_{\theta/2}(g^{m+1}(z))}, {0<\theta=3/4<1}) for {0<t<1} sufficiently small (so that {0<t^{\log(1/\lambda)/\log\sigma}<3/8}). In other words, we verified (CFa) for {Dg(g^m(z))=B}, i.e., when we iterate {g} once to get from {g^m(z)\in W} to {g^{m+1}(z)\in V}.

At this point, it is clear that the verification of (CFa) along the entire {g}-orbit of {z} is complete by “bootstrap”, i.e., by replacing {z} by {g^{m+1}(z)\in V} in the previous argument, etc. ad infinitum.

Once we checked (CFa), let’s quickly verify (CFb). Again, by symmetry, it suffices to check that the vectors inside {\mathcal{C}^u_{1/2}(z)} are expanded. By scaling, it suffices to consider vectors {v=v^u+b(1,0)\in\mathcal{C}^u_{1/2}(z)}, i.e., {a=1} and {|b|\leq1/2}. In this case, we have {v=(1+b,y^u)}, so that {A^n(v)=(\lambda^n(1+b), \sigma^n y^u)} for {0\leq n\leq m}.

For sake of simplicity, let us assume that {\log(1/t)/\log\sigma=N_t} is an integer, i.e., {N_t\in\mathbb{N}}. Then,

  • If {N_t<m}, we have that {Dg^{N_t+1}(z)=A^{N_t+1}} and

    \displaystyle \begin{array}{rcl} \|Dg^{N_t+1}(z)(v)\|&=&\|A^{N_t+1}(v)\|=\max\{\lambda^{N_t}(1+b),\sigma^{N_t}|y^u|\}\geq \sigma^{N_t}|y^u| \\ &=& (1/t)|y^u| \geq (2/\sqrt{t})\geq (4/3)(1/\sqrt{t})\|v\| \end{array}

    Here, we used that {|y^u|\geq 2\sqrt{t}} and {\|v\|=|1+b|\leq 3/2} for {|b|\leq1/2}.

  • If {N_t=m}, we have {Dg^{N_t+1}(z)=BA^{N_t}} and

    \displaystyle BA^{N_t}(v)=\frac{\lambda^m(1+b)}{x^s}(1,0)+ \left(\sigma^m y^u - \frac{\lambda^m(1+b)}{x^s}\right)v^u

    and, since, {|y^u|,|x^s|\geq 2\sqrt{t}}, {1/2\leq \|v\|=|1+b|\leq 2} (as {|b|\leq1/2}) and {0<\lambda<1}, we obtain that

    \displaystyle \begin{array}{rcl} \|BA^{N_t}(v)\|&\geq& \left|\sigma^{N_t} y^u - \frac{\lambda^{N_t}(1+b)}{x^s}\right|\cdot |y^u| \\ &\geq & \left(\frac{2}{\sqrt{t}} - \frac{\lambda^{N_t}}{\sqrt{t}}\right)\cdot 2\sqrt{t} \geq (4/3)\|v\|. \end{array}

In any event, we deduce that {Dg^{N_t+1}(z)} uniformly expands (by a factor {\geq 4/3>1}) vectors of {\mathcal{C}^u_{1/2}(z)} for {z\in V}, that is, the verification of item (CFb) is complete.

This finishes the verification of the conditions of the invariance cone field criterion and, therefore, the sketch of Proposition 5.

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