Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
Alex Blumenthal Jinxin Xue Lai-Sang Young


Limiting curves for   polynomial adic systems.[Supported by the RFBR (grant 14-01-00373)]
    A. R. MinabutdinovNational Research University Higher School of Economics, Department of Applied Mathematics and Business Informatics, St.Petersburg, Russia, e-mail: 
    December 30, 2023
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We consider a large class of 2D area-preserving diffeomorphisms that are
not uniformly hyperbolic but have strong hyperbolicity properties on large regions of
their phase spaces. A prime example is the standard map. Lower bounds for 
Lyapunov exponents of such systems are very hard to estimate, due to the potential
switching of “stable" and “unstable" directions. This paper shows 
that with the addition of (very) small random perturbations, one obtains with
relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.






§ INTRODUCTION


A signature of chaotic behavior in dynamical systems is sensitive dependence on
initial conditions. Mathematically, this is captured by the positivity of Lyapunov
exponents: a differentiable map F of a Riemannian manifold M is said to have 
a positive Lyapunov exponent (LE) at x ∈ M if dF^n_x grows exponentially fast 
with n. This paper is about volume-preserving diffeomorphisms, and 
we are interested in behaviors that occur on positive Lebesgue measure sets. 
Though the study of chaotic systems occupies a good part of smooth ergodic theory, the hypothesis of positive LE is extremely difficult to verify
when one is handed a concrete map defined by a specific equation – except
where the map possesses a continuous family of invariant cones. 

An example that has come to symbolize the enormity of the challenge is the standard map, a mapping 
Φ=Φ_L of the 2-torus given by

    Φ (I, θ) = (I + L sinθ,  θ + I + L sinθ)

where both coordinates I, θ are taken modulo 2π and 
L ∈ is a parameter. For L ≫ 1, the map Φ_L has strong expansion 
and contraction, their directions separated by clearly defined invariant cones 
on most of the phase space – except on two narrow strips near θ = ±π/2
on which vectors are rotated violating cone preservation. 
As the areas of these “critical regions" tend to zero as L →∞, one might
expect LE to be positive, but this problem has remained unresolved:
no one has been able to prove, or disprove, the positivity of Lyapunov exponents 
for Φ_L for any one L, however large, 
 in spite of considerable
effort by leading researchers. The best result known <cit.> is that the LE of Φ_L is positive on sets of Hausdorff dimension 2 (which are very far from having positive Lebesgue measure). The presence of elliptic islands, which has been shown
for a residual set of parameters <cit.>, confirms that the obstructions to proving the positivity
of LE are real.


In this paper, we propose that this problem can be more tractable if one accepts
that dynamical systems are inherently noisy. We show,
for a class of 2D maps F that includes the standard map, that by adding a 
very small, independent random perturbation at each step, the resulting 
maps have a positive LE that correctly reflects the rate 
of expansion of F – provided that F has sufficiently large expansion
to begin with. More precisely, if dF∼ L,  L ≫ 1, on a large portion of the phase space, then random perturbations of size O(e^-L^2-ε) are sufficient for
guaranteeing a LE ∼log L. 

Our proofs for these results, which are very short compared to previous works
on establishing nonuniform hyperbolicity for deterministic maps 
(e.g. <cit.>)
are based on the following idea: We view the random process as a Markov chain 
on the projective bundle of the manifold on which the random maps act, and 
represent LE as an integral. Decomposing this integral into a “good part" and
a “bad part", we estimate the first leveraging the strong hyperbolicity of the 
unperturbed map, and obtain a lower bound for the second provided
the stationary measure is not overly concentrated in certain “bad regions". 
We then use
a large enough random perturbation to make sure that the stationary measure
is sufficiently diffused.

We expect that with more work, this method can be extended both to higher
dimensions and to situations where conditions on the unperturbed map are 
relaxed. 


Relation to existing results. Closest to the present work are the unpublished results of Carleson and Spencer <cit.>, 
who showed for very carefully selected parameters L ≫ 1 of the standard map that 
LE are positive when the map's derivatives are randomly perturbed. 
For comparison, our first result applies to all L ≫ 1 with a slightly larger
perturbation than in <cit.>, and our second result
assumes additionally a finite condition on a finite set; we avoid the rather delicate
parameter selection by perturbing the maps themselves, not just
their derivatives. 

Parameter selections similar to
those in <cit.> were used – without random perturbations – 
to prove the positivity of LE 
for the Hénon maps <cit.>, quasi-periodic cocycles <cit.>, and rank-one attractors <cit.>, building on earlier 
techniques in 1D, see e.g. <cit.>. 
See also <cit.>, which estimates LE from below
for Schrödinger cocycles over the standard map. 
Relying on random perturbation alone – without parameter deletion – are <cit.>, which contains results analogous to ours in 1D, and
<cit.>, which applied random rotations to twist maps.
We mention also <cit.>, which uses hyperbolic toral automorphisms
in lieu of random perturbations.

Farther from our setting, the literature on LE is vast. Instead of
endeavoring to give reasonable citation of individual papers, let us mention
several categories of results in the literature that have attracted much attention,
together with a small sample of results in each.  Furstenberg's work <cit.> in the early 60's initiated extensive research on criteria for the LE of random matrix products to be distinct (see e.g. <cit.>).
Similar ideas were exploited to study LE of cocycles over hyperbolic
and partially hyperbolic systems (see e.g. <cit.>), with a generalization to deterministic maps <cit.>. Unlike the results in the first two paragraphs, these results 
do not give quantitative estimates; they assert only that LE are simple, or nonzero. 

We mention as well the formula of Herman <cit.> and the related work <cit.>, which use subharmonicity to estimate Lyapunov exponents, and the substantial body of work on 1D Schrödinger operators (e.g.  <cit.>). We also note the C^1 genericity of zero Lyapunov exponents of volume-preserving surface diffeomorphisms away from Anosov <cit.> and its higher-dimensional analogue <cit.>. Finally, we acknowledge results on the continuity or stability of LE, as in, e.g., <cit.>. 



This paper is organized as follows: We first state and prove two results in a relatively
simple setting: Theorem <ref>, which contains the core idea of this paper,  is proved in
Sections 3 and 4, while Theorem <ref>, which
shows how perturbation size can be decreased if some mild 
conditions are assumed, is proved in Section 5. 
We also describe a slightly more general setting
which includes the standard map, and observe in Section 6 that the proofs
given earlier in fact apply, exactly as written, to this broader setting.






§ RESULTS AND REMARKS




 §.§ Statement of results


We let ψ : 𝕊^1 → be a C^3 function for which the following hold:


  (H1) C_ψ' = {x̂∈𝕊^1 : ψ'(x̂) = 0} and C_ψ” = {ẑ∈𝕊^1 : ψ”(ẑ) = 0} have finite cardinality.

  (H2) min_x̂∈ C_ψ' |ψ”(x̂)| > 0 and min_ẑ∈ C_ψ” |ψ”'(ẑ)| > 0.

For L > 1 and a ∈ [0,1), we define 

    f=f_L,a: 𝕊^1 →        
    f(x) = L ψ(x) + a .

Let 𝕋^2 = 𝕊^1 ×𝕊^1 be the 2-torus. 
The deterministic map to be perturbed is 

    F=F_L,a: 𝕋^2 →𝕋^2          
    F(x,y) = ( [ f(x) - y   (mod 1);                  x ])  .

We have abused notation slightly in Eq (<ref>): 
We have made sense of f(x)-y
by viewing y ∈𝕊^1 as belonging in [0,1), and have
written “z (mod 1)" instead of
π(z) where π:ℝ→𝕊^1 ≅ℝ/ℤ is the usual
projection.
Observe that F is an area-preserving diffeomorphism of 𝕋^2.

We consider compositions of random maps

    F^n_ = F_ω_n∘⋯∘ F_ω_1       n=1,2, …,

where 

    F_ω = F ∘ S_ω ,      
    S_ω(x,y)  = ( x + ω  (mod 1), y) ,

and the sequence =(ω_1, ω_2, …) is chosen i.i.d. with respect 
to the uniform distribution ν^ on [- , ] for some > 0. Thus our sample space can be 
written as Ω = [- , ]^,
equipped with the probability =(ν^)^.

Throughout, we let Leb denote Lebesgue measure on ^2. 


Assume ψ obeys (H1),(H2), and fix a ∈ [0,1).  Then 


  (a) for every L>0 and >0, 

    λ_1^ = lim_n →∞1/nlog (d F_^n)_(x,y)

exists and is independent of (x, y, ) for every (x,y) ∈𝕋^2 and -a.e. ∈Ω;

  (b) given , ∈̱(0,1), there is a constant C = C_α, β > 0 such 
that for all L, where L is sufficiently large (depending on ψ, , )̱
and  ≥ L^- CL^1 - β, we have

    λ_1^≥log L   .






Theorem <ref>
assumes no information whatsoever on dynamical properties of F beyond its
definition in Eq (<ref>).
Our next result shows, under some minimal, easily checkable, condition on
the first iterates of F, that the bound above on λ^_1
continues to hold for a significantly smaller . Let 𝒩_c(C_ψ') denote the c-neighborhood of C_ψ' in 𝕊^1. We formulate the following condition
on f=f_L,a:

(H3)(c)  For any x̂, x̂' ∈ C_ψ', we have that f x̂ - x̂' 
(mod 1) ∉𝒩_c(C_ψ') .

Observe that for L large, the set of a for which (H3)(c) is satisfied
tends to 1 as c → 0.

 Let ψ be as above, and fix an arbitrary 
c_0>0. Then given , ∈̱(0,1), there is a constant C = C_α, β > 0 such that for all L, a, where

– L is sufficiently large (depending on ψ,c_0,, $̱), 

–a ∈[0,1)is chosen so thatf = f_L, asatisfies  (H3)(c_0), and

–≥L^- CL^2 - β,


then we have

    λ_1^≥log L   .
 A slight extension
Letψ: 𝕊^1 →be as above. ForL > 0anda ∈[0,1), 
we writef_0 = f_ψ, L, a = L ψ+ a, and forε> 0define𝒰_ε, L(f_0) = { f : 𝕊^1 → such that  f-f_0_C^3 < L ε} .We letC'_fandC”_fdenote the zeros off'andf”. Below, (H3)(c)is to be
read withC'_fin the place ofC'_ψ.
We writeF_f(x,y) = (f(x) -y , x)forf ∈𝒰_ε, L(f_0). 






















 Let ψ : 𝕊^1 → satisfy (H1), (H2) as before. For a ∈ [0,1) and L >1, let f_0 be as defined above. Then there exists ε>0 sufficiently
small so that 

(1) Theorems <ref> and <ref> hold for F = F_f
for all L > 0 sufficiently large and f ∈𝒰_ε, L(f_0); 

(2) L depends only on ψ as before but a in Theorem <ref> 
depends on f.

The Chirikov standard map is defined as follows: a parameterL > 0is fixed, and the map(I, θ) ↦(I̅, θ̅), sending[0,2 π)^2into itself, is defined by

    I̅   = I + 2 π L sinθ  , 
    θ̅   = θ + I̅ = θ + I + 2 π L sinθ  ,

where both coordinatesI, θare taken modulo2 π. 

 Let L be sufficiently large. Then:

  *  Theorem <ref> holds for the standard map.

  *  If additionally the map f(x) = L sin(2 π x) + 2 x satisfies (H3)(c) for some c > 0, then Theorem <ref> holds for the standard map for this value of L.

Theorem <ref> and Corollary <ref> are proved in Section 6.  All discussions prior to Section 6
pertain to the setting described at the beginning of this section.

 §.§ Remarks
 Remark 1. Uniform hyperbolicity on large but non-invariant regions 
of the phase space. An important property of the deterministic mapFis that 
cone fields can be defined on all of𝕋^2in such a way
that they are preserved bydF_(x,y)for(x,y)in a large but non-invariant 
region in^2. For example, letC_1/5={v=(v_x, v_y): |v_y/v_x| ≤1/5}.
Then for(x,y) ∉{|f'| < 10}, which by (H1) and (H2) is comprised 
of a finite number of very narrow vertical strips in𝕋^2forLlarge, 
one checks easily thatdF_(x,y)mapsC_1/5intoC_1/5, and expands vectors in these cones
uniformly. It is just as easy to see that this cone invariance property 
cannot be extended across the strips in{|f'| < 10}, and thatFis not uniformly hyperbolic. 

These “bad regions" where the invariant cone property fails
shrink in size asLincreases. More precisely, 
letK_1 > 1be such that|ψ'(x)| ≥K_1^-1 d(x, C_ψ');
that such aK_1exists follows from (H1), (H2) in Sect. 2.1. It is easy to check that
for anyη∈(0,1),d(x, C_ψ') ≥K_1/ L^1 - η     
|f'(x)| ≥ L^η ,and this strong expansion in thex-direction 
is reflected indF_(x,y)for anyy.

We must stress, however, that regardless of how small these “bad regions" are, the positivity of Lyapunov exponents is not guaranteed for the deterministic mapF– except for the Lebesgue measure zero set of orbits that never venture into 
 these regions. In general, tangent vectors that have expanded in the good regions
can be rotated into contracting directions when the orbit visits a bad region. 
This is how elliptic islands are formed.

 Remark 2. Interpretation of condition (H3). We have seen that visiting neighborhoods ofV_x̂ := {x=x̂}forx̂ ∈C'_ψcan lead to
a loss in hyperbolicity, yet
at the same time it is unavoidable that the “typical" orbit will visit these “bad regions".
Intuitively, it is logical to expect the situation to improve if we do not permit orbits to visit these bad regions two iterates in a row – except
that such a condition is impossible to arrange: sinceF(V_x̂')= {y=x̂'}, 
it follows thatF(V_x̂')meetsV_x̂for everyx̂, x̂' ∈C'_ψ. 
In Theorem <ref>, 
we assert that in the case of random maps, to reduce the size ofit suffices to impose the condition that no orbit can be inC'_ψ×𝕊^1for  three consecutive iterates. That is to say, supposeF(x_i,y_i)=(x_i+1,y_i+1),  i=1,2,…. Ifx_i, x_i+1 ∈C'_ψ, 
thenx_i+2must stay away fromC'_ψ. This is a rephrasing of (H3). 
Such a condition is both realizable and checkable, as it involves
only a finite number of iterates for a finite set of points.

 Remark 3. Potential improvements. 
Condition (H3) suggests that one may be able to shrinkfurther by imposing
similar conditions on one or two more iterates
ofF. Such conditions will cause the combinatorics in Section 5 to be more involved, 
and since our, which is∼L^-L^2-, is already extremely small for largeL,
we will not pursue these possibilities here.





§ PRELIMINARIES


The results of this section apply to allL, >0unless otherwise stated.



 §.§ Relevant Markov chains


Our random maps system{F^n_}_n ≥1can be seen as a 
time-homogeneous Markov chain𝒳 := {(x_n, y_n)}given by 

    (x_n, y_n) = F^n_(x_0, y_0) = F_ω_n(x_n - 1, y_n - 1)   .

That is to say, for fixed, the transition probability starting from(x,y) ∈𝕋^2is

    P((x,y), A) = P^((x,y), A) = ν^{ω∈ [- , ] : F_ω(x,y) ∈ A }

for BorelA ⊂𝕋^2. We writeP^(k)((x,y), ·)(orP^(k)_(x,y)) for the correspondingk-step transition probability. It is easy to see that for this chain, Lebesgue measure is stationary,
meaning for any Borel setA ⊂𝕋^2,

    (A) = ∫ P((x,y), A)   d (x,y)   .


Ergodicity of this chain is easy and we dispose of it quickly.


Lebesgue measure is ergodic.
 For any (x,y) ∈𝕋^2 and ω_1, ω_2 ∈ [-,], 

    F_ω_2∘ F_ω_1(x,y) = F ∘ F ∘ S'_ω_1, - ω_2(x,y)

where S'_ω, ω'(x,y) = ( x + ω  (mod  1), y + ω' (mod  1) ). That is to say, P^(2)_(x,y) is supported on
the set F^2([x-, x+] × [y-,y+]), on which it is equivalent to
Lebesgue measure. From this one deduces immediately 
that (i) every ergodic stationary measure of 𝒳 = {(x_n, y_n)} has a density, 
and (ii) all nearby points in 𝕋^2
are in the same ergodic component. Thus there can be at most one ergodic 
component.


Part (a) of Theorem <ref> follows immediately from Lemma <ref> together with the
Multiplicative Ergodic Theorem for random maps.


Next we introduce a Markov chain𝒳̂on^2, the projective bundle over^2. Associatingθ∈^1 ≅[0, π)with the unit vectoru_θ= (cosθ, sinθ),F_ωinduces a mappingF̂_ω: ^2 →^2defined byF̂_ω (x,y,θ) = (F_ω(x,y), 
θ')          u_θ' = 
±(dF_ω)_(x,y) u_θ/(dF_ω)_(x,y) u_θ .Here±is chosen to ensure thatθ' ∈[0,π). The Markov chain𝒳̂ : = {(x_n,y_n,θ_n)}is then defined by(x_n, y_n,θ_n) = F̂_ω_n(x_n-1, y_n-1, θ_n-1) .We writeP̂for its transition operator,P̂^(n)for then-step transition transition operator, and use Leb to denote also Lebesgue measure on^2.

For any stationary probability measureμ̂of the Markov chain(x_n,y_n,θ_n), defineλ(μ̂) = ∫log(dF_ω)_(x,y) u_θ dμ̂(x,y,θ) 
 dν^ (ω) .For any stationary probability measure μ̂ of
the Markov chain 𝒳̂, we have
λ^_1 ≥λ(μ̂) .
 By the additivity of the 
cocycle (x,y,θ) ↦logd(F_ω)_(x,y) u_θ, we have, for any n ∈, 

    λ (μ̂)     =    ∫1/nlog(dF_^n)_(x,y) u_θ dμ̂(x,y,θ)  d (ν^)^n()
       ≤   ∫1/nlog(dF_^n)_(x,y) d  (x,y)  d (ν^)^n() .

That μ̂ projects to Lebesgue measure on ^2 is used in passing from 
the first to the second line, and the latter converges to 
λ^_1 as n →∞ by the Multiplicative Ergodic Theorem.


Thus to prove part (b) of Theorem <ref>, it suffices to prove thatλ(μ̂) ≥logLfor someμ̂. Uniqueness ofμ̂is not required. On the other hand, once we have
shown thatλ^_1>0, it will follow that there can be at most
oneμ̂withλ(μ̂)>0. 
Details are left to the reader.

We remark also that while Theorems <ref>–<ref> hold for arbitrarily large values 
of, we will treat only the case≤1/2, leaving the very minor
modifications needed for the> 1/2case to the reader. 

Finally, we will omit from time to time the notation “(mod 1)" when the meaning 
is obvious, e.g. instead of the technically correct
but cumbersomef(x+ω) - y ,
we will writef(x+ω) - y.  








 §.§ A 3-step transition


In anticipation for later use, we compute here the transition probabilitiesP̂^(3)((x,y,θ), ·), also denotedP̂^(3)_(x,y,θ).
Let(x_0, y_0, θ_0) ∈^2be fixed.
We defineH=H^(3)_(x_0, y_0, θ_0) : [-,]^3 →^2byH(ω_1, ω_2, ω_3) = F̂_ω_3∘F̂_ω_2∘F̂_ω_1 (x_0, y_0, θ_0) .ThenP̂^(3)_(x_0,y_0,θ_0) = H_*((ν^)^3), the pushforward of(ν^)^3on[-,]^3byH.  Write(x_i, y_i,θ_i) = F̂_ω_i (x_i-1, y_i-1,θ_i-1),i = 1,2,3. 

 Let ∈ (0, 1/2]. Let (x_0, y_0, θ_0) ∈^2 be fixed, and let
H= H^(3)_(x_0, y_0, θ_0) be as above. Then 

  (i)
    dH(ω_1, ω_2, ω_3) =  sin^2 (θ_3) tan^2(θ_2) tan^2(θ_1) f”(x_0 + ω_1)   ;

  (ii) assuming θ_0 π/2, we have that
dH  0 on V where V ⊂
[-,]^3 is an open and dense set having full Lebesgue measure in [-,]^3;

  (iii)H is at most #(C_ψ”)-to-one, i.e., no point in ^2 has more than # C_ψ” preimages.
 The projectivized map F̂_ω can be written as

    F̂_ω(x,y,θ)=(f(x+ω)-y ,x+ω, arctan1/f'(x+ω)-tanθ)   ,

where arctan is chosen to take values in [0,π).


(i) It is convenient to write k_i=tanθ_i, so that k_i + 1 = (f'(y_i + 1) - k_i)^-1. Note as well that x_i + 1 = f(y_i + 1) - y_i. Then

    dx_3∧ dy_3∧ dθ_3 
    =     (f'(y_3)dy_3-dy_2)∧ dy_3∧(∂θ_3/∂ y_3dy_3+∂θ_3/∂ k_2dk_2
    )
    
    =   -dy_2∧ dy_3∧(∂θ_3/∂ k_2dk_2)
    
    =     -dy_2∧(dω_3+f'(y_2)dy_2-dy_1)∧(∂θ_3/∂ k_2)(∂ k_2/∂ y_2dy_2+∂ k_2/∂ k_1dk_1)
    
    =    -dy_2∧ d(ω_3-y_1)∧(∂θ_3/∂ k_2∂ k_2/∂ k_1dk_1)
    
    =    -(dω_2+f'(y_1)dy_1)∧ d(ω_3-y_1)∧(∂θ_3/∂ k_2∂ k_2/∂ k_1∂ k_1/∂ y_1 dy_1)
    
    =   -dω_2∧ dω_3∧(∂θ_3/∂ k_2∂ k_2/∂ k_1∂ k_1/∂ y_1 dω_1).

It remains to compute the parenthetical term. The second two partial derivatives are straightforward. The first partial derivative is computed by taking the partial derivative of the formula θ_3 = f'(y_3) - k_2 with respect to k_2 on both sides. 
We obtain as a result

    ∂θ_3/∂ k_2∂ k_2/∂ k_1∂ k_1/∂ y_1= - sin^2θ_3tan^2θ_2 tan^2θ_1 f”(x_0+ω_1)   .

(ii) For x ∈ [0,1) and θ∈ [0,π) ∖{π/2}, define U(x, θ) = {ω∈ [- , ] : f'(x + ω) - tanθ≠ 0}. Note that U(x, θ) has full Lebesgue measure in [- , ] by (H1). We define

    V = {(ω_1, ω_2, ω_3) ∈ [- , ]^3 :     ω_1 ∈ U(x_0, θ_0), ω_2 ∈ U(x_1, θ_1), 
       ω_3 ∈ U(x_2, θ_2) ,  and  f”(x_0 + ω_1) ≠ 0}  .

By (H1) and Fubini's Theorem, V has full measure in [- , ]^3, and it is clearly open 
and dense. To show dH  0, 
we need θ_i  0 for i=1,2,3 on V. This  follows from the fact that for 
θ_i-1≠π/2, if ω_i ∈ U(x_i-1, θ_i-1) then θ_i ≠ 0, π/2.


(iii) Given (x_3, y_3, θ_3), we solve for (ω_1, ω_2, ω_3)
so that H(ω_1, ω_2, ω_3)=(x_3, y_3, θ_3). Letting
(x_i, y_i, θ_i),  i = 1,2, be the intermediate images, we note that 
y_2 is uniquely determined by x_3 = f(y_3) - y_2, θ_2 is determined
by θ_3 =  f'(y_3) - tanθ_2, as is θ_1 once θ_2 and y_2 are fixed. This in turn determines f'(x_0 + ω_1), but here uniqueness of
solutions breaks down.











Let ω_1^(i)∈ [- , ], i=1, …, n, give the required value of f'(x_0+ω_1^(i)). We observe that each ω_1^(i) determines uniquely  
y_1^(i) = x_0 + ω_1^(i), x_1^(i)=f(y_1^(i))-y_0,
ω_2^(i) = y_2-x_1^(i), x_2^(i) = f(y_2) - y_1^(i), and
finally ω_3^(i) = y_3 - x_2^(i). 
Thus the number of H-preimages of any one point in ^2 cannot exceed n.
Finally, we have n ≤ 2 for  small, and n ≤# (C_ψ”) for  as large as 1/2.
 For any stationary probability μ̂ of 𝒳̂,
we have μ̂(^2 ×{π/2}) =0, and for any 
(x_0,y_0,θ_0) with θ_0 π/2 and any (x_3, y_3,θ_3) ∈^2, 
the density of P̂^(3)_(x_0,y_0,θ_0) at (x_3, y_3,θ_3)
is given by

    1/(2)^3(∑_ω_1 ∈ℰ(x_3, y_3,θ_3)1/|f”(x_0 + ω_1)|)   1/ρ(x_3,y_3,θ_3)

where
ℰ(x_3, y_3,θ_3) = {ω_1: ∃ω_2, ω_3
 H(ω_1, ω_2, ω_3) = (x_3, y_3,θ_3)}
and
ρ(x,y,θ) = sin^2 (θ) [f'(f(y) - x)  (f'(y) - θ) - 1 ]^2 .
 To show μ̂(×{π/2}) = 0, it suffices to show that 
given any x ∈ [0,1) and any θ∈ [0,π), 
ν^{ω∈ [- , ] : f'(x + ω) = tanθ} = 0, and 
that is true because C”_ψ is finite by (H1). The formula in (<ref>) 
follows immediately
from the proof of Lemma <ref>, upon expressing tan^2(θ_2) tan^2(θ_1)
in terms of (x_3,y_3,θ_3) as was done in the proof of Lemma <ref>(iii).



§ PROOF OF THEOREM <REF>


The idea of our proof is as follows: Letμ̂be any stationary probability
of the Markov chain𝒳̂.
To estimate the integral inλ(μ̂), we need 
to know the distribution ofμ̂in theθ-direction. 
Given that the mapsF_ωare strongly uniformly hyperbolic on a large part of the phase space 
with expanding directions well aligned with thex-axis  (see Remark 1),
one can expect that underdF^N_for largeN,μ̂will be pushed toward a neighborhood of{θ=0}on much of^2,
and that is consistent withλ^_1 ≈logL. This reasoning, however, 
is predicated onμ̂not being concentrated, or stuck, on very small sets far away 
from{θ≈0}, a scenario not immediately ruled out
 as the densities of transition probabilities are not bounded.

We address this issue directly by proving
 in Lemma <ref> an  a priori bound 
on the extent to whichμ̂-measure can be concentrated on (arbitrary) small sets. 
This bound is used in Lemma <ref> to estimate theμ̂-measure of the
set in^2not yet attracted to{θ=0}inNsteps. The rest of the proof
consists of checking that these bounds are adequate for our purposes.




In the rest of the proof, letμ̂be an arbitrary invariant 
probability measure of𝒳̂.

 Let A ⊂{θ∈ [π/4, 3 π/4]} be a Borel subset of ^2. Then for L large enough,

    μ̂(A) ≤Ĉ/L^1/4(1 + 1/^3 L^2(A) )   ,

for all ∈ (0, 1/2], where Ĉ > 0 is a constant independent of L, or A.
 
By the stationarity of μ̂, we have, for every Borel set 
A ⊂^2,

    μ̂(A) =  ∫_^2P̂^(3)_(x_0, y_0, θ_0)(A) 
      d μ̂(x_0, y_0, θ_0) .

Our plan is to decompose this integral into a main term and “error terms", depending
on properties of the density of P̂^(3)_(x_0, y_0, θ_0). The decomposition
is slightly different depending on whether ≤ L^-1/2 or ≥ L^-1/2.

 The case ≤ L^-1/2. Let K_2 ≥ 1 be such that |ψ”(x)| ≥ K_2^-1 d(x,C”_ψ); such
a K_2 exists by (H1),(H2). Define B” = {(x, y) : d(x,C_ψ”) ≤ 2K_2 L^- 1/2}.
Then splitting the right side of (<ref>) into

    ∫_B”× [0, π)  +  ∫_^2 ∖ (B”× [0, π)) ,

we see that the first integral is ≤(B”) ≤4K_2 M_2/√(L) where M_2 = # C_ψ”. As for (x_0,y_0) ∉B”, since |f”(x_0+ω)| ≥ L^1/2, the density of 
P̂^(3)_(x_0, y_0, θ_0) is ≤ [(2)^3 M_2^-1 L^1/2ρ]^-1
by Corollary <ref>.

To bound the second integral in (<ref>), we need to consider the zeros of ρ. As A ⊂^2 × [π/4, 3 π/4], 
we have sin^2(θ_3) ≥ 1/2. The form of ρ in Corollary <ref> prompts us to decompose A into 

A = (A ∩Ĝ) ∪ (A ∖Ĝ)
 
where Ĝ = G × [0,π) and

G = {(x,y) : d(y, C'_ψ) > K_1 L^-1/2,  d(f(y)-x, C'_ψ) ≥ K_1 L^-1/2} .

Then on Ĝ∩ A, we have
ρ≥1/2 (1/2 L)^2 for L sufficiently large. This gives
∫_^2 ∖ (B”× [0, π))P̂^(3)_(x_0, y_0, θ_0)(A ∩Ĝ)   d μ̂≤C/^3 √(L)1/L^2 (A) .

Finally, by the invariance of μ̂,
∫_^2 ∖ (B”× [0, π))P̂^(3)_(x_0, y_0, θ_0)(A ∖Ĝ)   d μ̂ ≤ μ̂(A ∖Ĝ) = 
(^2 ∖ G) .

We claim that this is ≲ L^-1/2. Clearly, Leb{d(y, C'_ψ) ≤ K_1 L^-1/2})
≈ L^-1/2. As for the second condition,
{y: f(y) ∈ (z-K_1 L^-1/2, z+ K_1 L^-1/2)}
= {y: ψ(y) ∈ (z'-K_1 L^-3/2, z'+ K_1 L^-3/2)}
which in the worst case has Lebesgue measure ≲ L^-3/4 by (H1), (H2).

 The case ≥ L^-1/2. Here we let B̃” = {(x,y) : d(x,C_ψ”) ≤ K_2 L^- 3/4}, and decompose the right side of (<ref>) into

    ∫ (P̂^(3)_(x_0, y_0, θ_0))_1(A)   d μ̂ +  ∫ (P̂^(3)_(x_0, y_0, θ_0))_2(A)   d μ̂
 
where, in the notation in Sect. 3.2, 


    (P̂^(3)_(x_0, y_0, θ_0))_1 = 
    H_* ((ν^)^ 3|_{x_0 + ω_1 ∈B̃”})     
    (P̂^(3)_(x_0, y_0, θ_0))_2 = 
    H_* ((ν^)^ 3|_{x_0 + ω_1 ∉B̃”})  .

Then the first integral is bounded above by
sup_x_0 ∈𝕊^1ν^{ω_1 ∈B̃” - x_0}≲^-1(B̃”) ≤ Const · L^- 1/4 ,

while the density of (P̂^(3)_(x_0, y_0, θ_0))_2 is ≤  [(2)^3 M_2^-1 
L^1/4·ρ(x_3,y_3,θ_3)]^-1. The second integral is 
treated as in the case of ≤ L^-1/2. 

As discussed above, we now proceed to estimate
 the Lebesgue measure of the set that remains far
away from{θ=0}afterNsteps, whereNis arbitrary for  now.
For fixed= (ω_1, …, ω_N ), we write(x_i,y_i) = F^i_ (x_0,y_0)for1 ≤i ≤N, and defineG_N=G_N(ω_1, …, ω_N )by

    G_N = {(x_0, y_0) ∈^2 :      d(x_i + ω_i + 1, C_ψ') ≥ K_1 L^-1 +  for all  0 ≤ i ≤ N-1  } .

We remark that for(x_0, y_0) ∈G_N, the orbitF^i_ (x_0,y_0), i ≤N,
passes through uniformly hyperbolic regions of^2, where invariant
cones are preserved and|f'(x_i + ω_i + 1)| ≥L^βfor eachi<N; see Remark 1 in Section 2.
We further defineĜ_N = { (x_0,y_0,θ_0) : (x_0,y_0) ∈G_N}.

 Let β>0 be given. We assume 
L is sufficiently large (depending on $̱).
Then for anyN ∈ℕ,∈ (0, 1/2]andω_1, …, ω_N  ∈ [- , ],μ̂(Ĝ_N ∩{|tanθ_N| > 1}) ≤Ĉ/L^1/4(1+ 1/^3 L^2+β N)  .For (x_0,y_0) ∈ G_N, consider the singular value decomposition of 
(dF_^N)_(x_0, y_0). Let ϑ^-_0 denote the angle corresponding to the most contracted direction of (dF_^N)_(x_0, y_0) and ϑ_N^- its
image under (dF_^N)_(x_0, y_0), and let σ > 1 > σ^-1 denote
the singular values of (dF_^N)_(x_0, y_0). A straightforward computation gives

    1/2 L^β≤ |tanϑ^-_0|, |tanϑ_N^-|    and   σ≥( 1/3 L^)^N   .

It follows immediately that for fixed (x_0,y_0), 
{θ_0 : |tanθ_N| >1}⊂ [π/4, 3π/4] and
{θ_0 : |tanθ_N| >1} <  L^-β N .
 
Applying Lemma <ref> with A = Ĝ_N ∩{ | tanθ_N| > 1},
we obtain the asserted bound.

By the stationarity ofμ̂, it is true for anyNthatλ(μ̂) = ∫( ∫ (dF_ω_N + 1)_(x_N, y_N) u_θ_N d (F̂_ω_N∘…∘F̂_ω_1)_* μ̂)
dν^(ω_1) ⋯ d ν^(ω_N + 1)   .We have chosen to estimateλ(μ̂)one sample path at a time because  
we have information from Lemma <ref> on(F̂_ω_N∘…∘F̂_ω_1)_*μ̂for each sequenceω_1, …, ω_N.


Let α, ∈̱(0,1). Then, there are constants C = C_α, β > 0
and C' = C_α, β' > 0 such that for any L sufficiently large, we have the following. Let N = ⌊ C' L^1 - ⌋, ∈ [ L^- C L^1 - β, 1/2], and fix arbitrary ω_1, ⋯, ω_N + 1∈ [- , ]. Then,

    I  :=  ∫_^2log (d F_ω_N + 1)_(x_N, y_N) u_θ_N  d μ̂(x_0,y_0, θ_0)  ≥ log L   .


Integrating (<ref>) over(ω_1, …, ω_N + 1)givesλ(μ̂) ≥log L. 
Asλ^_1 ≥λ(μ̂), part (b) of Theorem <ref> follows
immediately from this proposition.

 The number N will be determined in the course of the proof, 
and L will be enlarged a finite number of times as we go along. 
As usual, we will split I, the integral in (<ref>), to one on a good and 
a bad set.
The good set is essentially the one in Lemma <ref>, with an additional condition on 
(x_N,y_N), where dF will be evaluated. Let 

    G^*_N = {(x_0, y_0) ∈ G_N : d(x_N + ω_N + 1, C_ψ') ≥ K_1 m }  ,

where m > 0 is a small parameter to be specified later. 
As before, we let Ĝ^*_N = G^*_N × [0,2π).
Then 𝒢:= Ĝ^*_N ∩{|tanθ_N| ≤ 1} is the good set; 
on 𝒢, the integrand in (<ref>) is ≥log( m L/4). Elsewhere we use the worst
 lower bound - log (2 ψ'_C_0 L). Altogether we have
 
    I ≥ log (1/4 m L) - logm ψ' L^2/2 μ̂(ℬ)  ,

where 

    ℬ = ^2 ∖𝒢 =
    (^2 ∖Ĝ^*_N) ∪ (Ĝ^*_N 
    ∩{|tanθ_N| > 1}) .

We now bound μ̂(ℬ). First, 

    μ̂(^2 ∖Ĝ^*_N) = 1- (G^*_N) ≤ K_1 M_1 (m + NL^-1 + )   ,

where M_1 = # C_ψ'. Letting N= ⌊ C'L^1-⌋ and m=C'=p/4K_1M_1
where p is a small number to be determined,
we obtain μ̂(^2 ∖Ĝ^*_N) ≤1/2 p. 
From Lemma <ref>, 

    μ̂(Ĝ^*_N ∩{|tanθ_N| > 1}) ≤Ĉ/L^1/4(1 +  1/( L^1/3Ṉ)^3) .

For N as above and  in the designated range (with C =  ̱/3 C'), 
the right side of (<ref>) is easily made < 1/2 p 
by taking L large, so we have μ̂(ℬ) ≤ p. 
Plugging into (<ref>), we see that

I  ≥ (1 - 2p) log L - {log p,  plog p  } .

Setting p = 1/4 (1 - ) and taking L large enough, one ensures that
I > log L.



§ PROOF OF THEOREM <REF>


We now show that with the additional assumption (H3), the same result holds
for≥ L^- C L^-2+.



 §.§ Proof of theorem modulo main proposition


As the idea of the proof of Theorem <ref> closely parallels that of Proposition <ref>, it is useful to recapitulate the main ideas:

  1. The main Lyapunov exponent estimate is carried on the subset
{(x_0,y_0,θ_0): (x_0,y_0) ∈ G_N,
|tanθ_N|<1} of ^2, where G_N
consists of points whose orbits stay 
 ≳ L^-1+ away from C'_ψ×𝕊^1 in their first N iterates.

  2. Since Leb(G^c_N)∼ NL^-1+, we must take N ≲ L^1-.

  3. By the uniform hyperbolicity of F^N_ on G_N, Leb{|tanθ_N|>1}∼ L^-cN.

  4. For μ̂{|tanθ_N|>1} to be small, we must have 1/^3 L^-cN≪ 1 (Lemma <ref>).
 
Items 2–4 together suggest that we require > L^-1/3 cN≥ 
L^-c'L^-1+, and we checked that for thisthe proof goes through.


The proof of Theorem <ref> we now present differs from the above in the following way:
The setG_N, which plays the same role as in Theorem <ref>, will be different.
It will satisfy 

  (A) Leb(G^c_N)∼ NL^-2+, and

  (B) the composite map dF^N_ is uniformly hyperbolic on G_N.

The idea is as follows: To decrease, we must increaseN, while keeping
the setG^c_Nsmall. This can be done by allowing the random orbit to come closer toC_ψ' ×𝕊^1, but with that, one cannot expect uniform hyperbolicity
in each of the firstNiterations, so we require only (B). This is the main difference
between Theorems <ref> and <ref>.
OnceG_Nis properly identified and properties (A) and (B) are proved, the rest of
the proof follows that of Theorem <ref>: Property (A) permits us to takeN ∼ L^2-in item 2, and item 3 is valid by Property (B).
Item 4 is general and therefore unchanged, leading
to the conclusion that it suffices to assume > L^-c'L^-2+.
As the arguments follow those in Theorem <ref> verbatim modulo the bounds above and 
accompanying constants, we will not repeat
the proof. The rest of this section is focused on
producingG_Nwith the required properties.

 It is assumed from here on that (H3)(c_0)holds,
andL, aandare as in Theorem <ref>. Having proved Theorem <ref>, 
we may assume≤ L^-1. In light of the discussion above,ω_1, ⋯, ω_N , ω_N + 1∈ [-,]will be fixed throughout, and(x_i,y_i)=F^i_(x_0,y_0)as before. 

 Definition of G_N. For arbitraryNwe defineG_Nto be

    G_N = {(x_0, y_0) ∈^2 : 
    (a)     for all  0 ≤ i ≤ N -1,  
            (i)  d(x_i + ω_i + 1 , C_ψ') ≥ K_1 L^-2 +  , 
            (ii)  d(x_i + ω_i + 1 , C_ψ')· d(x_i+1 + ω_i + 2 , C_ψ') ≥ K_1^2 L^-2+ /̱ 2 , 
    
    (b)     d(x_0 + ω_1, C_ψ'),  d(x_N-1 + ω_N, C_ψ')≥ p/(16M_1)
    }

whereM_1 = # C_ψ'andp=p(α)is a small number to be determined. 
Notice that (a)(i)  implies only|f'(x_i+ω_i+1)| ≥ L^-1 + ,
not enough to guarantee expansion in the horizontal direction. We remark also that even though (a)(ii)
implies|f'(x_i + ω_i + 1) f'(x_i+1 + ω_i + 2)| ≥ L^/̱ 2,
hyperbolicity does not follow without control of the  angles of the vectors involved. 

 There exists C_2 ≥ 1 such that for all N, 
(G_N^c) ≤ C_2 NL^-2+ + p/4 .

Let 

    A_1    = {x ∈ [0,1) : d(x, C_ψ') ≥ K_1 L^-2 + }  , 
    
    A_2    = {(x, y) ∈^2 : x ∈ A_1,  and  d(x, C_ψ') · d(f x - y, C_ψ') ≥ K_1^2 L^-2 + /̱2}  .

We begin by estimating (A_2). Note that (A_1^c) ≤ 2 M_1 K_1 L^-2 +, and for each fixed x ∈ A_1, 

    {y ∈ [0,1) : d(f x - y, C_ψ') < K_1^2 L^-2 + /̱2/d(x, C_ψ')}≤2 M_1 K_1^2 L^-2 + /̱2/d(x, C_ψ')  ,

hence

    A_2^c    ≤ A_1^c + ∫_x ∈ A_12 M_1 K_1^2 L^-2 + /̱2/d(x, C_ψ') dx   .

Let ĉ = 1/2min{ d(x̂,x̂')  : x̂, x̂' ∈ C_ψ', x̂≠x̂'} . We split the integral above into ∫_d(x, C_ψ') > ĉ+ ∫_K_1 L^-2 + ≤ d(x, C_ψ') ≤ĉ. The first one is bounded from above by 2 M_1 K_1^2 ĉ^-1 L^-2 + /̱2, and the second by

    4 M_1^2 K_1^2 L^-2 + /̱2∫_ K_1 L^-2 + ^ĉdu/u≤ 4 (2 - )̱ M_1^2 K_1^2 L^-2 + /̱2log L     ,

(having used that - log K_1 and logĉ are < 0) and so on taking L large enough so that L^/̱2≥log L, it follows that (A_2^c) ≤ C_2 L^-2 +, where C_2 = C_2, ψ depends on ψ alone.

Let G̃_N be equal to G_N with condition (b) removed. Then
G̃_N = ⋂_i = 0^N-1  (F_^i)^-1(  A_2 - (ω_i + 1, 0) ) ,

so (G̃_N) ≥ 1 - C_2 N L^-2 +. The rest is obvious.
 For any N ≥ 2, 
(dF^N_)_(x_0, y_0) is hyperbolic on G_N with the following uniform bounds:
The larger singular value σ_1 of (dF^N_)_(x_0, y_0) satisfies

    σ_1(  (dF^N_)_(x_0, y_0)) ≥ L^/15 N   ,

and if ϑ_0^- ∈ [0,π) denotes the most contracting direction of (dF^N_)_(x_0, y_0) and ϑ_N^- ∈ [0,π) its image, then 

    |ϑ_0^- - π/2|, |ϑ_N^- - π/2| ≤ L^-   .

The bulk of the work in the proof of Theorem <ref> goes into proving this proposition.



 §.§ Proof of Property (B) modulo technical estimates


 Letc =c_ψ≪ c_0wherec_0is as in (H3); we stipulate additionally thatc ≤ p / 16 M_1, wherep=  p_αandM_1are as before. First we introduce the following symbolic encoding of^2. LetB = 𝒩_√(c/L)( C'_ψ) ×𝕊^1,    
I = 𝒩_c( C'_ψ) ×𝕊^1 ∖ B, 
     G = ^2 ∖ (B ∪ I) .To each(x_0, y_0) ∈^2we associate a symbolic sequence(x_0,y_0)  ↦ W̅ = W_N-1⋯ W_1 W_0 ∈{B, I, G}^N ,where(x_i + ω_i + 1, y_i) ∈ W_i. We will refer to any symbolic sequence of length≥ 1, 
e.g.V̅ = GBBG, as a  word, and use Len(V̅)to denote the length
ofV̅, i.e., the number of letters it contains. We also writeG^kas shorthand for a word consisting
ofkcopies ofG. Notice that symbolic sequences are to be read from right
to left.

The following is a direct consequence of (H3). 

 Assume that < L^-1. Let (x_0,y_0) ∈^2 be such that (x_0 + ω_1, y_0) ∈ B ∪ I and (x_1 + ω_2, y_1) ∈ B.
Then (x_2 + ω_3, y_2) ∈ G. 
 Let x̂_0, x̂_1 ∈ C'_ψ (possibly x̂_0 = x̂_1) 
 be such that d(x_0,x̂_0 )< c and d(x_1,x̂_1) < √(c/L).
Since (f(x̂_1) - x̂_0) ∉𝒩_c_0( C'_ψ)
by (H3), it suffices to show |x_2 - (f(x̂_1) - x̂_0) | ≪ c_0:

    |x_2 - (f(x̂_1) - x̂_0) |     =    
    |(f(x_1 + ω_2)-y_1) - (f(x̂_1) - x̂_0) |
       ≤    | f(x_1 + ω_2) -f(x̂_1)| + d(y_1, x̂_0) .
 
To see that this is ≪ c_0, observe that for large L, we have

|f(x_1 + ω_2) - f(x̂_1)| < 1/2 L ψ”(√(c/L) + 
L^-1)^2 < ψ” c ,

and d(y_1, x̂_0) = d(x_0+ω_1, x̂_0) < 2c.






















Next we apply Lemma <ref> to put constraints on the set of all possible wordsW̅associated with(x_0, y_0) ∈ G_N.


Let W̅ be associated with  (x_0, y_0) ∈ G_N. Then
W̅ must have the following form:

    W̅ = G^k_MV̅_M G^k_M-1V̅_M-1⋯ G^k_1V̅_1 G^k_0  ,

where M ≥ 0, k_0, k_1, ⋯, k_M ≥ 1, and if M > 0, then each
 V̅_i is one of the words in
𝒱 =  {B, BB,   B I^k B, I^k B, I^k, B I^k  k ≥ 1 } .

The sequence W̅ starts and ends with G by the definition of G_N and the stipulation that c ≤ p / (16 M_1); thus a decomposition of the form (<ref>) is obtained with words {V̅_i}_i = 1^M formed from the letters {I, B}. To show that the words {V̅_i}_i = 1^M must be of the proscribed form, observe that

  * BB occurs only as a subword of GBBG;

  * BI only occurs as a subword of GBI;

  * IB only occurs as a subword of IBG.

Each of these constraints follows from Lemma <ref>; for the third,
G is the only letter that can precede IB. It follows from the last two bullets that all 
the Is must be consecutive, and B can appear at most twice.


With respect to the representation in (<ref>), we view eachV̅_ias representing
an excursion away from the “good region"G. In what follows, 
we will show thatG_Nand (H3) are chosen so that
for(x_0,y_0) ∈ G_N, vectors are not rotated by too much during these excursions, 
and hyperbolicity is restored with each visit toG. 
To prove this, we introduce the following cones in tangent space:

    𝒞_n = 𝒞(L^-1 + /̱4)   ,     𝒞_1 = 𝒞(1)   ,   and   𝒞_w = 𝒞(L^1 - /̱4)   ,

where𝒞(s)refers to the cone of vectors whose slopes have absolute value≤ s. The lettersn, wstand for `narrow' and `wide', respectively.

Let(x_0,y_0) ∈ G_Nand suppose for somemandl,{(x_m+i-1 + ω_m+i, y_m+i-1)}_i=1^lcorresponds to the wordV̅ = V_l ⋯ V_1 ∈𝒱. 
To simplify notation, we write(x̃_i, ỹ_i)= (x_m+i-1 + ω_m+i, y_m+i-1)         
dF̃^l = dF_(x̃_l, ỹ_l)∘⋯∘ dF_(x̃_1, ỹ_1) .Let {(x̃_i, ỹ_i)}_i = 1^l and V̅ = V_l ⋯ V_1∈𝒱
be as above. Then

dF̃^l (𝒞_n)  ⊂𝒞_w ,
  (dF̃^l) ^* (𝒞_n)  ⊂𝒞_w ,     min_u ∈𝒞_n,  u  = 1 dF̃^l u ≥ 1/2 L^/5 n_I (V̅)
where n_I(V̅) is the number of appearances of the letter I in the word V̅. 



We defer the proof of Proposition <ref> to the next subsection.



For (x_0, y_0) ∈ G_N, let W̅ be as in (<ref>). It is easy to check
that if (x_m + ω_m + 1, y_m) ∈ G, then

    (dF_ω_m + 1)_(x_m ,y_m)  (𝒞_w) ⊂𝒞_n
        min_u ∈_w, u = 1(dF_ω_m + 1)_(x_m ,y_m)  u≥1/4 L^/̱4  .

Applying (<ref>) and Proposition <ref> alternately, we obtain

    (dF^N_)_(x_0, y_0) (_w) ⊂_n   .

Identical considerations for the adjoint yield the cones relation (dF^N_)_(x_0, y_0)^* _w ⊂_n. We now use the following elementary fact from linear algebra: if M is a 2 × 2 real matrix with distinct real eigenvalues η_1 >  η_2 and corresponding eigenvectors v_1, v_2 ∈^2, and if 𝒞 is any closed convex cone with nonempty interior for which M 𝒞⊂𝒞, then v_1 ∈𝒞. 

We therefore conclude that the maximal expanding direction ϑ_0^+ for (dF^N_)_(x_0, y_0) and its image ϑ_N^+ both belong to 𝒞_n. The estimates for ϑ_0^-, ϑ_N^- now follow on recalling that ϑ_0^- = ϑ_0^+ + π/2   (mod π), ϑ_N^- = ϑ_N^+ + π/2   (mod π).

It remains to compute σ_1 ( (dF^N_)_(x_0, y_0)). From 
(<ref>) and the derivative bound in Proposition <ref> gives

    min_u ∈_w, u = 1(dF^N_)_(x_0, y_0) u≥ L^/5( (k_0 - 1) + (k_1 - 1) + ⋯ + (k_M-1 - 1) + k_M + ∑_i = 1^M (n_I(V̅_i) + 1) )

As there cannot be more than two copies of B in each V̅∈𝒱, 
we have

    n_I(V̅) + 1/(V̅) + 1≥1/3  ,

and the asserted bound follows.



 §.§ Proof of Proposition <ref>


Cones relations for adjoints are identical to those of the original, and so are omitted: hereafter we work exclusively with the original (unadjointed) derivatives. We will continue to use the notation in Proposition <ref>. Additionally, 
 in each of the assertions below, ifdF̃^lis applied to the cone, thenminrefers to the minimum
taken over all unit vectorsu ∈.




The proof consists of enumerating all cases ofV̅∈𝒱.
We group the estimates as follows:

 
  (a) For V̅ = I:   dF̃ (_1) ⊂_1   and   min dF̃ u≥1/2 K_1 √(c)√(L)≫ L^1/4.

  (b) For V̅ = B:   dF̃ (_n) ⊂_w   and   min dF̃ u≥1/2.


The next group consists of two-letter words the treatment of which will rely on 
condition (a)(ii) in the definition ofG_N. 

 
  (c) For V̅ = BB:   dF̃^2(_n) ⊂_w   and   mindF̃^2u ≥ L^/̱3.

  (d) For V̅ = BI:   dF̃^2(_1) ⊂_w   and   mindF̃^2 u≥min{1/2 K_1 √(c)√(L) , L^/̱3}≥ L^/̱5.

  (e) For V̅ = IB:   d F̃^2 (_n) ⊂_1   and   mindF̃^2u ≥ L^/̱3.


This leaves us with the following most problematic case:
 
  (f) For V̅ = BIB:   dF̃^3 (_n) ⊂_w   and   mindF̃^3 u≥ L^/̱5 .

We go over the following checklist:




  * V̅ = B or BB was covered by (b) and (c); total growth on _n is ≥1/2.

For k ≥ 1, 

  * V̅ = I^k follows from (a); total growth on _n is ≥ L^k /4≫ L^/5 k.

  * V̅ = I^k B = I^k-1(IB) follows from concatenating (e) and (a); total growth on _n is 
≥ L^(k-1) / 4· L^/̱3≫ L^/5 k.

  * V̅ = B I^k = (BI)I^k-1 follows from concatenating (a) and (d); total growth on _n is 
≥ L^/̱5· L^(k-1)/4≫ L^/5 k.

Lastly, 

  * V̅ = BIB follows from (f); total growth on _n is ≥ L^/̱5,
and

  *  for k ≥ 2, V̅ = BI^kB = (BI) I^k-2 (IB) follows by concatenating (e), followed
by (a) then
 (d); total growth on _n is ≥ L^/̱5· L^(k-2) /4· L^/̱3≫ L^/5 k.

This completes the proof.


Lemma <ref> is easy and left to the reader; it is a straightforward application of the formulae

    tanθ_ 1 = 1/f'(x̃_1) - tanθ_0  ,    d F̃ u_θ = √((f'(x̃_1) cosθ_0 - sinθ_0)^2 + cos^2 θ_0)  ,

whereθ_1 ∈ [0, π)denotes the angle of the image vectord F̃ u_θ_0.

Below we letKbe such that|f'| ≤ KL.


We write u = u_θ_0 and θ_1, θ_2 ∈ [0,2 π) for the angles of the images dF̃ u, dF̃^2u respectively. Throughout, we use the following `two step' formulae:

    tanθ_2 = f'(x̃_1) - tanθ_0/f'(x̃_1) f'(x̃_2) - f'(x̃_2) tanθ_0 - 1  , 
     dF̃^2 u_θ≥ |( f'(x̃_1)  f'(x̃_2) - 1) cosθ_0| - | f'(x̃_2) sinθ_0|

The estimate |f'(x̃_1) f'(x̃_2)| ≥ L^/̱ 2 (condition (a)(ii) in the definition
of G_N) will be used repeatedly throughout.



We first handle the vector growth estimates. For (c) and (e), as u = u_θ_0∈_n, the right side of (<ref>) is
≥1/2 L^/̱2 - 2 K L^/̱4≫ L^/̱3.








For (d) we break into the cases 

(d.i) |f'(x̃_2)| ≥ L^/̱4 and 

(d.ii) |f'(x̃_2)| < L^/̱4. 

 In case (d.i), by (a) we have that u_θ_1∈_1 and d F_(x̃_1, ỹ_1) u_θ_0≥1/2 K_1 √(c)√(L). Thus |tanθ_2| ≤ 2 L^- /̱4≪ 1 and dF_(x̃_2, ỹ_2) u_θ_1≥1/2 L^/̱4≫ 1, completing the proof. In case (d.ii), the right side of (<ref>) is

    ≥1/√(2) (L^/̱2 - 1) - 1/√(2) L^/̱4≫ L^/̱3  .


We now check the cones relations for (c) – (e). For (c), 

    |tanθ_2| ≤|f'(x̃_1)| + |tanθ_0|/|f'(x̃_1) f'(x̃_2) | - |f'(x̃_2) tanθ_0| - 1≤K L + L^-1 +/̱4 /L^/̱2 - K L^/̱4 - 1≤ 2 K L^1 - /̱2≪ L^1 - /̱4  ,

so that u_θ_2∈_w as advertised. The case (d.i) has already been
treated. For (d.ii), the same bound as in (c) gives

    |tanθ_2| ≤K L + 1 /L^/̱2 - L^/̱4 - 1≤  2 K L^1 - /̱2≪ L^1 - /̱4  ,

hence u_θ_2∈_w.

For (e) we again distinguish the cases 

(e.i) |f'(x̃_1)| ≥ L^/̱ 4 and 

(e.ii) |f'(x̃_1)| < L^/̱ 4. 

 In case (e.i), one easily checks that dF_(x̃_1, ỹ_1) (_n) ⊂_1 and then dF_(x̃_2, ỹ_2) (_1) ⊂_1 by (a). In case (e.ii) we compute directly that

    |tanθ_2| ≤|f'(x_1)| + |tanθ_0|/|f'(x_1) f'(x_2) | - |f'(x_ 2) tanθ_0| - 1≤L^/̱4 + L^-1 + /̱4/L^/̱2 - K L^/̱4 - 1≪ 1   ,

hence u_θ_2∈_1.

We let u = u_θ_0∈_n (i.e. |tanθ_0| ≤ L^-1 + /̱4) and write θ_1, θ_2, θ_3 ∈ [0,π) for the angles associated to the subsequent images of u. We break into two cases: 

(I) |f'(x̃_3)| ≥ |f'(x̃_1)| and 

(II) |f'(x̃_3)| < |f'(x̃_1)|.


In case (I), we compute

    |tanθ_2 |≤|f'(x̃_1)| + |tanθ_0|/|f'(x̃_1) f'(x̃_2)| - |f'(x̃_2) tanθ_0| - 1≤2 |f'(x̃_1)|/L^/̱2 - 2 K L^/̱4 - 1≤ 4 |f'(x̃_1)| L^- /̱2  ,

having used that |f'(x̃_1)| ≥ L^-1 + and |tanθ_0| ≤ L^-1 + /̱4 in the second inequality. Now,

    |tanθ_3| ≤1/|f'(x̃_3)| - |tanθ_2|≤1/|f'(x̃_1)| - 4 |f'(x̃_1)| L^- /̱2≤2/|f'(x̃_1)|≤ 2 L^1 - ≪ L^1 - /̱4  .

In case (II), we use

    |tanθ_1| ≤1/|f'(x̃_1)| - |tanθ_0|≤1/|f'(x̃_3)| - L^-1 + /̱4≤2/|f'(x̃_3)|  ,

again using that |f'(x̃_3)| ≥ L^-1 +, and then

    |tanθ_3|    ≤   |f'(x̃_2)| + |tanθ_1|/|f'(x̃_2) f'(x̃_3)| - |f'(x̃_3) tanθ_1| - 1
       ≤   K L + 2 |f'(x̃_3)|^-1/L^/̱2 - 3≤K L + 2 L^1 - /L^/̱2 - 3≤ 2 K L^1 - /̱2≪ L^1 - /̱4  .


For vector growth, observe that from (e) we have
dF̃^2(_n) ⊂_1 and mindF̃^2 u≥ L^/̱3.
So, if |f'(x̃_3)| ≥ L^/̱12 then 

    dF_(x̃_3, ỹ_3) u_θ_2≥1/√(2) (L^/̱12 -1 ) ≫ 1   .
 
Conversely, if |f'(x̃_3)| < L^/̱12 then we can use the crude estimate (dF_(x̃, ỹ))^-1≤√(|f'(x̃)|^2 + 1) applied to (x̃, ỹ) = (x̃_3, ỹ_3), yielding

    (dF_(x̃_3, ỹ_3))^-1≤√(L^2/̱12 + 1)≤ 2 L^/̱12  ,

hence dF̃^3 u≥1/2 L^/̱3 - /̱12 = 1/2 L^/̱4≫ L^/̱5, completing the proof.



§ THE STANDARD MAP



Letψandf_0 = f_ψ, L, a = L ψ + abe as defined in Sect. 2.1.


There exists ε > 0 and K_0 > 1, depending only on ψ, for which the following holds: for all L > 0 and f ∈𝒰_ε, L(f_0),

  (a)max{f'_C^0, f”_C^0,  f”'_C^0}≤ K_0 L,

  (b) The cardinalities of C'_f and C”_f are equal to those of f_0 (equivalently
those of ψ), 

  (c)min_x̂∈ C_f' |f”(x̂)|   ,  min_ẑ∈ C_f” |f”'(ẑ)| ≥ K_0^-1 L , and 

  (d)min_x̂, x̂' ∈ C'_f d(x̂,x̂' )  ,  min_ẑ, ẑ' ∈ C”_f 
d(ẑ, ẑ') ≥ K_0^-1

The proof is straightforward and is left to the reader.

























We claim – and leave it to the reader to check – that the proofs in Sections 3–5 (with C_f', C_f” replacing C_ψ', C_ψ”)
use only the form of the maps F= F_f as defined in Sect. 2.1,
and the four properties above. Thus they prove Theorem <ref> as well.

Under the (linear) coordinate change x = 1/2 πθ, y = 1/2 π(θ - I), the standard map conjugates to the map

    (x,y) ↦ (L sin (2 π x) + 2 x - y, x)

defined on ^2, with both coordinates taken modulo 1. This map is of the form F_f, with f(x) = f_0(x) +2  x and f_0 (x) : =L sin (2 π x); here a = 0 and ψ(x) = sin(2 π x). 
Let ε>0 be given by Theorem <ref> for this choice of ψ.
Then f clearly belongs in 𝒰_ε, L(f_0) for large
enough L. 
siam