An example of a T n endomorphism that is persistently singular and C 1 robustly transitive
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A PERSISTENTLY SINGULAR MAP OF T n THAT IS C ROBUSTLY TRANSITIVE.
Juan Carlos Morelli ∗ Universidad de La Rep´ublica. Facultad de Ingenieria. IMERL.Julio Herrera y Reissig 565. C.P. 11300.Montevideo, Uruguay. (Communicated by )
Abstract.
Consider the high dimensional torus T n and the set E of its en-domorphisms endowed with the C topology. A map F in E that is robustlytransitive and presents a non empty and persistent critical set is exhibited. Introduction.
Whenever we think about dynamical systems’ properties almostinevitably come to mind the concepts of stability and robustness . Loosely speaking,we can say that stability implies same dynamics for maps sufficiently close to eachother, and robustness implies the same behavior relative to a specifical property formaps sufficiently close to each other. These are both of most importance in thestudy of any dynamical system.This work in particular is centered in the study of robust transitivity, meaningby transitive the existence of a forward dense orbit of a point. This may seem atfirst sight as an unexciting topic since a fair amount of results concerning robusttransitivity are known. Nonetheless, the aimed class of maps, the singular endo-morphisms about which little to nothing is known; as well as taking on the highdimensional context are undoubtedly a fresh approach to the subject.To set ideas in order we list up the most relevant known results about the topic.We begin summing up the most studied case: robust transitivity of diffeomorphisms.The image provided by known results is fairly complete. Concerning surfaces, [13]shows that robust transitivity implies Anosov diffeomorphism and manifold T ;while in dim ( M ) = n manifolds, in [4] is proved that robust transitivity implies adominated splitting.Going further, next comes robust transitivity of regular endomorphisms (not glob-ally but locally invertible). The image we have about these is somewhat less com-plete: we know that volume expanding is a necessary but not sufficient conditionfor C robust transitivity according to [10], they also give a sufficient condition forthe case of manifold T n .Carrying on, at last there is the least studied case, robust transitivity of singularmaps (non empty critical set). Until 2013 nothing had ever been written on thetopic. It was on that year when [3] showed the first example of a C transitive Mathematics Subject Classification.
Primary: 37C20; Secondary: 57R45, 57N16.
Key words and phrases.
Transitivity, singularity, stability, robustness, high dimension. singular map of T . The second example was given only in 2016 by [8], they show a C robustly transitive map of T with a persistent critical set. Nothing more thanthese two examples was known until that time.Even so, there have been recent further advances on robust transitivity of singularsurface endomorphisms: in 2018 [9] presented an example of a T whose robusttransitivity depends on the class of differentiabilty, and in 2019 [11] and [12] set the state of the art proving that partial hyperbolicity is a necesary condition, that theonly surfaces that support them are T and the Klein bottle, and that they belongto the homotopy class of a linear map with an eigenvalue of modulus larger thanone.Concerning the higher dimensional context, the only known result was given by[14] where he extended the result appearing in [9] to T n . The present article showsthat the example presented in [8] can also be extended to T n . It would be the sec-ond known study of persistently singular endomorphisms on manifolds of dimensionlarger that 2, as well as the first known examples of its type. The main results canbe stated as: Theorem 1.1.
Given n ≥ , there exists a persistently singular endomorphism of T n that is C robustly transitive. Sketch of the Construction.
Starting from a diagonal matrix with integercoefficients we construct and endomorphism of T n with a very strong unstabledirection and a blending region where the local dynamics determine the map tobe robustly transitive. Afterwards we perturb this map outside of the blendingregion to obtain a new map that presents a persistent critical set and whose robusttransitivity is unaffected, resulting in the sought map announced in the title.2. Preliminaries.
Some basic definitions are recalled at the beginning. The readeris assumed to be familiar with the concepts of real manifold and submanifold, atlas,chart, tangent vector and tangent space, differentiable map and differential of amap, etc. For more details about the contents of this section the reader might referitself to [5] or [6].Let M be a differentiable manifold of dimension m (without loss of generality,consider M to be compact, connected and without boundary) and f : M → M adifferentiable endomorphism. Definition 2.1.
The orbit of x ∈ M is O ( x ) = { f n ( x ) , n ∈ N } . Definition 2.2. f is transitive if there exists a point x ∈ M such that O ( x ) = M . Theorem 2.1. If f is continuous then are equivalent:1. f is transitive.2. For all U, V open sets in M , exists n ∈ N such that f n ( U ) ∩ V = ∅ .3. There exists a residual set R (countable intersection of open and dense sets)such that for all points x ∈ R : O ( x ) = M . Definition 2.3. f is C k - robustly transitive if there exists ε > U ( f,ε ) of f in the C k topology such that g is transitive for all g ∈ U f .2.1. Blenders.
A brief overview of the concept of a blender is given now. In mostsituations it is easy to think of blenders as higher dimensional horseshoes, or assets exhibiting the dynamics of a Smale’s horseshoe. Blenders force the robust
OBUST TRANSITIVITY OF SINGULAR ENDOMORPHISMS 3 intersection of topologically ’thin’ sets, giving rise to rich dynamics.According to [2], ”A blender is a compact hyperbolic set whose unstable set has dimensionstrictly less than one would predict by looking at its intersection withfamilies of submanifolds”.
They also provide with a prototipical example of a blender: Let R be a rectan-gle with two rectangles R and R lying inside, horizontally, and such that theirprojections onto the base of R overlap (Figure 1). Consider now a diffeomorphism f such that f ( R ) = f ( R ) = R . Then, Ω = T n ∈ N f − n ( R ) gives rise to a blender(Cantor) set for f . Observe that f admits a fixed point inside each of R and R ,and that all vertical segments between the projection of these points intersect Ω.Observe as well that this construction is robust in two senses: on the one hand, f can be slightly perturbed with persistance of the property. And on the other, thevertical segment can also be slightly perturbed and still intersect Ω. Figure 1.
A protoblender over R . Darker is f − ( R ).The alpha limit set α (Ω) contains the cartesian product of Ω with parallel lines tothe base of R , giving rise to a fractal set with analog properties to that of an unstablemanifold. Every close-to-vertical line in between the fixed points of f inside R ∪ R will cross Ω; hence, it behaves like a surface even when its topological dimension isone. For more insight on blenders and its applications the reader may go to [1].2.2. Iterated Function Systems.
Let F , G be two families of diffeomorphisms of M . Denote by F ◦ G := { f ◦ g/ f ∈ F , g ∈ G} ; and for k ∈ N denote F = { Id M } and F k +1 = F k ◦ F . Then, the set S ∞ k =0 F k has a semigroup structure that isdenoted by hFi + and said to be generated by F . Definition 2.4.
The action of the semigroup hFi + on M is called the iteratedfunction system associated with F . We denote it by IFS( F ). Definition 2.5.
For x ∈ M , the orbit of x by the action of the semigroup hFi + is hFi + ( x ) = { f ( x ) , f ∈ hFi + } . Definition 2.6.
A sequence { x n , n ∈ N } is a branch of an orbit of IFS( F ) if forevery n ∈ N there exists f n ∈ hFi + such that f n ( x n ) = x n +1 . JUAN C. MORELLI
Definition 2.7.
An IFS( F ) is minimal if for every x ∈ M the orbit hFi + ( x ) hasa branch that is dense on M .An IFS( F ) is C r robustly minimal if for every family ˆ F of C r perturbations of F and every x ∈ M the orbit h ˆ Fi + ( x ) has a branch that is dense on M .An IFS( F ) is C r strongly robustly minimal if for every family ˆ F of C r per-turbations of F and every x ∈ M the orbit h ˆ Fi + ( x ) has a branch that is dense on M and whenever f n ( x n ) = x n +1 then it is possible to assign to x n +1 the image ofˆ f n ( x n ) where ˆ f n is a C r perturbation of f n . Remark 2.1.
Strong robust minimality roughly means that a perturbation can beperformed at every iteration.The next theorem is crucial for the construction carried on in the article. Forthe proof, the reader may go to [7].
Theorem 2.2.
Every boundaryless compact manifold admits a pair of diffeomor-phisms that generate a C strongly robustly minimal IFS. If F = { g , g } is the family given by Theorem 2.2, then the following propertieshold:1. The diffeomorphism g admits a unique attracting fixed point a and a uniquerepelling fixed point r . Likewise, the diffeomorphism g admits a uniqueattracting fixed point a and a unique repelling fixed point r .2. There exists a blending region for IFS( F ) containing a and r on which g and g ◦ g are contractions.3. The maps g and g are not C close to the identity but can be constructedin such a way that the norms of the differntial maps k Dg k and k Dg k areeverywhere close to 1. Remark 2.2.
Since g and g are perturbations of a given Morse-Smale diffeomor-phism they can be chosen as C close to the identity as desired.3. A regular endomorphism f of T n . Having stated all the preliminary factsneeded to construct the example claimed in the title, we proceed to it now in twosteps. We firstly define an endomorphism f that is C robustly transitive andsecondly perturb f to a map F satisfying the claim of Theorem 1.1.3.1. Construction of f . Consider the n dimensional torus T n as the quotient IR n / [ − , n and endow it with the standard riemannian (euclidean) metric. Let b A ∈ M n ( ZZ ) be the diagonal matrix suggested below, with a large integer in thefirst entry and all of the other elements being 1, b A =
14 0 0 · · ·
00 1 0 · · · · · · · · · . (3.1)The matrix b A defines a regular endomorphism A on the torus defined by theformula A : T n → T n /A ( x , ..., x n ) = (14 x , x , ..., x n ). Remark 3.1. . OBUST TRANSITIVITY OF SINGULAR ENDOMORPHISMS 5
1. The construction could be carried on with any λ ∈ ZZ such that | λ | >>
1; thechoice of 14 is made in the sake of simplicity and for a better understandingof the contents to follow.2. Observe that A is a map modulo T n defined along this work.For the rest of the construction, consider a decomposition of the torus given by T n = S × T n − ; the map A becomes A : S × T n − → T n /A ( x, y ) = (14 x, y ).Define now the following subsets of the first factor S : K = (cid:2) − , (cid:3) , K = (cid:2) , (cid:3) and K = ( K ∪ K ). Take ε = and define the sets K ε = (cid:2) − − ε, + ε (cid:3) , K ε = (cid:2) − ε, + ε (cid:3) and K ε = ( K ε ∪ K ε ).Define next a smooth function u : IR → IR such that u | K = 1 and u | ( K ε ) c = 0 asshown in Figure 2. Observe that max x ∈ IR {| u ′ ( x ) |} exists. Figure 2.
Graph of u .Finally, let F = { g , g } be the family given by Theorem 2.2 for the secondfactor T n − , satisfying the properties claimed in the theorem and with the addi-tional requirement that both g and g are C close to the identity according tomax y ∈ T n − {|| y − g ( y ) || , || y − g ( y ) ||} ≤ {| u ′ |} .Define ˆ f : K ε × T n − → T n / ˆ f ( x, y ) = (cid:26) (14 x, g ( y )) if x ∈ K ε (14 x, g ( y )) if x ∈ K ε and extend ˆ f to f : S × T n − → T n /f ( x, y ) = (cid:26) u ( x ) . ˆ f ( x, y ) + (1 − u ( x )) .A ( x, y ) if x ∈ K ε A ( x, y ) if x / ∈ K ε (3.2) Remark 3.2.
Observe that the following hold:1. Taking ˆ f ( x, y ) = (14 x, ˆ f ( y )), then f ( x, y ) = (14 x, u ( x ) . ˆ f ( y ) + (1 − u ( x )) .y ).2. Since k Dg k and k Dg k are everywhere close to 1, then k D ˆ f k < f , k Id − ˆ f k ≤ {| u ′ |} .4. The restriction f | (cid:16) K × T n − (cid:17) = ˆ f .5. The restriction f | (cid:16) K ε × T n − (cid:17) c = A . JUAN C. MORELLI
It is straightforward to see that f has a strong dominant expanding directionalong the first coordinate. It follows that there exists a family of unstable cones for f in the direction of the canonical vector ~e . We make a pause here to check theexistence of the unstable cone field for f .Recall that for x ∈ M , we call cone of parameter a , index n − k and vertex x to C ua ( x ) = (cid:26) ( v , ..., v n ) ∈ T x M/ k ( v k +1 , ..., v n ) kk ( v , v , ..., v k ) k < a (cid:27) and that f admits an unstable cone of parameter a and vertex x ∈ M if thereexists C ua ( x ) ⊂ T x M such that D x f ( C ua ( x )) \ { } ⊂ C ua ( f ( x )). Lemma 3.1.
The map f defined by Equation 3.2 admits an unstable cone of pa-rameter , index n − and vertex ( x, y ) at every ( x, y ) ∈ T n . Proof:
The differential of f at ( x, y ) is given by D ( x,y ) f = (cid:18)
14 0 u ′ ( x ) . ( ˆ f ( y ) − y ) u ( x ) .D ˆ f + (1 − u ( x )) .y (cid:19) . Then for all vectors ( v , v ) of the tangent space of T n at ( x, y ) it is D ( x,y ) f ( v , v ) = v h u ′ ( x ) . ( ˆ f ( y ) − y ) i v + h u ( x ) .D ˆ f + (1 − u ( x )) .y i v ! . Consider now all vectors ( v , v ) in C u ( x, y ) and let ( w , w ) := Df ( x,y ) ( v , v ),we see that it is unstable by computing || w |||| w || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)h u ′ ( x ) . ( ˆ f ( y ) − y ) i v + h u ( x ) .D ˆ f + (1 − u ( x )) .y i v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | v | ≤≤ | u ′ ( x ) | . || ˆ f ( y ) − y ||
14 + ( | u ( x ) | . || D ˆ f || + | − u ( x ) | . || Id || ) . ≤ < . where in the first inequality we apply triangular and that ( v , v ) ∈ C u ( x, y ) and inthe second inequality we use: • ( v , v ) ∈ C u ( x, y ), • | u ′ ( x ) | . || ˆ f ( y ) − y || <
33 by Remark 3.2, • || D ˆ f || < • k Id k ≤ • max x ∈ IR {| u ( x ) | , | − u ( x ) |} ≤ (cid:3) Lemma 3.2.
For all v ∈ C u ( x, y ) holds that k D x f ( v ) k > k v k . Proof:
Let v = ( v , v ) ∈ C u ( x, y ): (cid:18) k D ( x,y ) f ( v , v ) k . k ( v , v ) k (cid:19) ≥ (14 . | v | ) . ( | v | + || v || ) ≥ (cid:18) (cid:16) || v ||| v | (cid:17) (cid:19) > > . (cid:3) Corollary 3.1.
For all curves γ such that for all t where γ is defined it holds that γ ′ ( t ) ∈ C u ( γ ( t )) , then the diameter easily satisfies diam ( f ( γ )) ≥ .diam ( γ ) . We go on next highlighting some of the other relevant dynamical features themap f possesses. All of them are straightforward to check: OBUST TRANSITIVITY OF SINGULAR ENDOMORPHISMS 7
Remark 3.3. K ε ⊂ [ − − ε, + ε ].2) f ( K ε × T n − ) ∩ f ( K ε × T n − ) ⊃ [ − − ε, + ε ] × T n − .3) The set K ε × T n − is a protoblender for f relative to [ − − ε, + ε ] × T n − .4) The points (0 , a ) and ( , a ) are saddle fixed points for f , and the points (0 , r )and ( , r ) are repelling fixed points for f .5) The local unstable manifold at (0 , a ) is W uloc (0 , a ) = ( − ε, ε ) × { a } .6) There exists an open ball B ⊂ T n − containing a such that the local stablemanifold at (0 , a ) is W sloc (0 , a ) = { } × B .We prove now that both the stable and unstable manifolds at (0 , a ) are densein T n . This will yield f is C robustly transitive. Lemma 3.3.
The unstable manifold W u (0 , a ) is dense in T n . Proof:
Let V = V × V be an open set in T n = S × T n − . We show next thatthere exists a point in the unstable local manifold W uloc (0 , a ) = ( − ε, ε ) × { a } witha forward iterate entering V in the future.Let f be f ( x, y ) = (14 x, f ( y )). Since f ( x ) = 14 x expands, there exists a naturalnumber k such that f k ( W uloc (0 , a )) ⊃ K × { f k (0) } . Observe that the preimage of f k +1 ( W uloc (0 , a )) also satisfies f − ( f k +1 ( W uloc (0 , a ))) ⊃ K × { f k +12 (0) } .Now, since IFS( F ) is minimal, there exists a branch of an orbit of f k +12 (0) thatintersects V at, let’s say, f k +1+ j (0). Considering the protoblender structure re-marked on item 3 above, the set T t ∈ N ∩ [0 ,j ] f − t ( S × { f k +1+ j (0) } ) projects onto S containing a finite segment of a Cantor set such that both K × T n − and K × T n − contain a full copy of the itinerary branch from f k +12 (0) up to f k +1+ j (0)where it enters V , by a preproduct of adequate open sets contained in S . Picka point p in K × { f k +12 (0) } of the infinitely many that satisfy f k +1+ j ( p ) ∩ V = ∅ and f − k − ( p ) ∈ W uloc (0 , a ) to get a point in the unstable manifold with a forwarditerate entering V . (cid:3) Lemma 3.4.
The stable manifold W u (0 , a ) is dense in T n . Proof:
Let V an open set in T n and W sloc (0 , a ) = { } × B for some appropriateopen set B in T n − containing a . Consider a point p ∈ V and a well definedcurve γ : ( − r, r ) → V / γ ( t ) = p + t. ~e . Decompose γ = ( γ , γ ) so γ is a constantfunction. Since for all t , γ ′ ( t ) = ~e , γ is a curve that travels inside the unstable conefield of f . Consequently there exists k ∈ N such that diam ( f k ( γ )) ≥ k .diam ( γ ) > f k ( γ ) ∩ K = ∅ . Pick a point q = ( q , q ) ∈ f k ( γ ) ∩ K . Again, since IFS( F ) isstrongly robustly minimal there exists a branch of the orbit hFi + ( q ) that enters B . At that iterate the length of the curve is such that it must cover all of S in thefirst coordinate, so there exists a point in V entering W sloc (0 , a ) in the future. (cid:3) Theorem 3.1.
The map f is robustly transitive. Proof:
According to Lemmas 3.3 and 3.4 there exists a point in T n with densestable and unstable manifolds. By a standard procedure of past and future iterationit holds that for all open sets U and V there exists a natural number k such that f k ( U ) ∩ V = ∅ which yields transitivity for f . Now, since Theorem 2.2 givesrobustness for IFS( F ) it is straightforward that Lemmas 3.4 and 3.3 hold in a U ∈ C neighborhood of f hence all maps in U are transitive. (cid:3) JUAN C. MORELLI A singular endomorphism F of T n . We procceed now to perform a per-turbation on the map f defined in the previous section in order to endow it withpersistent singularities without destroying its dynamical characteristics.We begin defining the singularities for any map h : M → M first; recall from thepreliminaries that M denotes a real manifold of dimension m . Definition 4.1.
We say that x ∈ M is a critical point or singularity for h if thedifferential map at x , D x h is not surjective.Observe that x is a singularity for h if and only if the rank of the jacobian matrixsatisfies rk ( D x h ) < m , if and only if the determinant det ( D x h ) = 0. Definition 4.2.
The critical set of h is S h = { x ∈ M/rk ( D x h ) < m } . Definition 4.3.
We say that h is a singular endomorphism if the critical set S h is non empty; and we say that h is a persistently singular endomorphism if there exists a neighborhood U h ∈ C of h in the C topology such that all g ∈ U h satisfy S g = ∅ .4.1. Construction of F . Sketch of the construction:
We choose a point not in K ε × T n − , set a ball around it where the introduction of the critical points takesplace; done in such a way that the critical set is persistent. Since the perturbationdoes not affect K × T n − , the transitivity of f is inherited by F .Let p = ( , , ..., , ) ∈ T n . Our goal is to define a ball of center p to perform aperturbation in order to obtain the map F we seek. To achieve this goal we need tofix a series of technical parameters; the choice to set all of them at the same timeand at the beginning of the construction is in expectance of avoiding darkness andof that it will be clear how they depend on each other.Start with r > B ( p,r ) ∩ (cid:0) K ε × T n − (cid:1) = ∅ , this is possiblesince p / ∈ (cid:0) K ε × T n − (cid:1) . Fix a second parameter θ such that 0 < θ < r and define asmooth ( C ∞ ) function ψ : IR → IR with an only critical point at , with ψ ( ) = 2and ψ ( x ) = 0 for all x in the complement of ( − θ, + θ ); and an axis of symmetryin the line x = as shown in Figure 3 (a) .Set finally a last parameter δ , with 0 < δ < θ verifying the following condition:since the derivative of ψ is bounded once θ has been fixed, name the bound as m ψ := m ψ ( θ ) = max x ∈ IR {| ψ ′ ( x ) |} and impose on δ that 8 .m ψ .r.δ < δ , consider another smooth function ϕ : IR → IR such that: • ϕ ′ is as in Figure 3 (b), • | ϕ ′ ( x ) | ≤ x ∈ IR , • ϕ ( ) = 0, ϕ ′ ( ) = , ϕ ′ ( + δ ) = 1, ϕ ′ ( + δ ) = − , • ϕ ( x ) = 0 for all x / ∈ [ − δ , + δ ]. Remark 4.1. max {| ϕ ( x ) | : x ∈ IR } ≤ δ. We are now in condition to define a perturbation of f in the direction of the lastcoordinate ~e n that depends on r , θ and δ which by simplicity we call only F and isdefined at x = ( x , ..., x n ) as F r,θ,δ : T n → T n /F ( x ) = ( f ( x ) if x / ∈ B ( p,r ) A ( x ) − ϕ ( x n ) .ψ (cid:16)P n − j =1 x j (cid:17) . ~e n if x ∈ B ( p,r ) . (4.1) OBUST TRANSITIVITY OF SINGULAR ENDOMORPHISMS 9 (a)(b)
Figure 3.
Graphs of ψ and ϕ ′ Remark 4.2.
1. For all x / ∈ B ( p, δ ) it holds that F ( x ) = f ( x ).2. For all x / ∈ K ε × T n − it holds that f ( x ) = A ( x ).To make the reading easier we will denote ϕ ( x n ) as ϕ and ψ (cid:16)P n − j =1 x j (cid:17) as ψ omitting the evaluations appearing on the definition. Lemma 4.1.
The endomorphism F defined by Equation 4.1 is persistently singular. Proof:
Start computing the differential D x F at x = ( x , ..., x n ) ∈ B ( p,r ) to get D x F =
14 0 · · · · · · · · · − .x .ϕ.ψ ′ − .x .ϕ.ψ ′ · · · − .x n − .ϕ.ψ ′ − ϕ ′ .ψ . (4.2)Since the critical set of F is defined as S F = { x ∈ T n /det ( D x F ) = 0 } , Equation4.2 provides det ( D x f ) = 14 · (1 − ϕ ′ .ψ ). In turn, S F = { x ∈ T n / − ϕ ′ .ψ = 0 } . Notice that S F is not empty since p = ( , , ..., , ) ∈ S F . To prove that S F is per-sistent, consider the points q = ( , , ..., , + δ ) and q = ( , , ..., , + δ ) bothin B ( p,r ) . Evaluate determinants to obtain det ( D q F ) = 35 and det ( D q F ) = − U ( F, ) ∈ C , every g ∈ U F satisfies S g = ∅ . (cid:3) We turn now to the last step of the construction where we show that F is C robustly transitive. To prove it, observe first that Lemma 3.3 holds for F auto-matically. If we prove that Lemma 3.4 also holds for F , then we can apply thesame reasoning of Theorem 3.1 to F to have the result. Notice that for Lemma3.4 to hold for F we only need to show that F admits an unstable cone C u ( x )at every point x ∈ B ( p,r ) satisfying that for all curves γ with γ ′ ∈ C u ( x ) then diam ( F ( γ )) > .diam ( γ ). Lemma 4.2.
For all x ∈ B ( p,r ) it holds that C u ( x ) is an unstable cone for F . Before moving on to the proof, we will denote as ˜ v = ( v , v , ..., v n − ) whenever v = ( v , v , ..., v n ). Proof:
By Equation 4.2 we have for all v = ( v , v , ..., v n ) ∈ C u ( x ) : D x F ( v ) = (14 .v , v , ..., v n − , − . h ˜ x, ˜ v i .ϕ.ψ ′ + v n . (1 − ϕ ′ .ψ )) . Call u ( u , .., u n ) := D x F ( v ) and perform calculations, we have k u , ..., u n k| u | = k ( v , ..., v n − , − . h ˜ x, ˜ v i .ϕ.ψ ′ + v n . (1 − ϕ ′ .ψ ) k| .v | ≤≤ k v , ..., v n − k| .v | + 2 . k ˜ x k . k ˜ v k . | ϕ | . | ψ ′ || .v | + | − ϕ ′ .ψ | . | v n || .v | <
314 +2 .r. (cid:18) (cid:19) .δ.m ψ + 914 < • v ∈ C u ( x ), • k ˜ x k ≤ k x k < r , • k ˜ v k| .v | ≤ | v | + k v ,...,v n − k| .v | ≤ + , • | ϕ | < δ , • | ψ ′ | < m ψ , • | − ϕ ′ .ψ | ≤ − ≤ ϕ ′ ≤ ≤ ψ ≤ .m ψ .r.δ <
29 imposed over δ . (cid:3) Lemma 4.3.
For all x ∈ B ( p,r ) and all v ∈ C u ( x ) it holds that k D x F ( v ) k > k v k . Let v = ( v , v ) ∈ C u ( x ): (cid:18) k D ( x,y ) F ( v , v ) k . k ( v , v ) k (cid:19) ≥ .v . ( v + || v || ) ≥ . (cid:16) || v || | v | (cid:17) > > . (cid:3) Lemma 4.4.
The map F defined by Equation 4.1 is C robustly transitive. Proof:
From Lemmas 4.2 and 4.3 we conclude that Lemma 3.3 holds for F . It wasalready mentioned that Lemma 3.4 holds for F . Consequently, Theorem 3.1 holdsfor F . (cid:3) Theorem 4.1.
There exists a persistently singular endomorphism F : T n → T n that is C robustly transitive. OBUST TRANSITIVITY OF SINGULAR ENDOMORPHISMS 11
Proof:
Define U ∈ C an open neighborhood of F where Lemma 4.1 holds and U ∈ C an open neighborhood of F where Lemma 4.4 holds. Then, all maps in U F = U ∩ U are C robustly transitive and have nonempty critical set. (cid:3) Acknowledgements.
The author would like to give thanks to Dr. Jorge Iglesiasfor fruitful conversations regarding this problem and to Dr. Roberto Markarian forhis generous attitude towards the author’s work.
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