Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $\mu_{\max}$ supported on a periodic orbit that maximizes the integral of $\phi$ among all $f$ invariant Borel probability measures.


Introduction
A relatively new field of study, ergodic optimization has displayed under a new point of view several distinct problems in dynamical systems, and enjoyed the benefits of allying techniques from optimization theory and ergodic theory to address them. Its usual setup is a dynamical system f : X → X, where X is a topological space, and a potential function φ : X → R, and the prototypical problem in the field is to determine, among all f invariant Borel probability measures M inv (f ), if there exists measures that maximize the functional P φ : M inv (f ) → R, P φ (µ) = φdµ and to further characterize these maximizing measures in term of their support.
Several problems can be put under this context, like finding Lyapunov exponents, action minimizing solutions to Lagrangian systems and the zero temperature limits of Gibbs equilibrium states in thermodynamical formalism. Some of the first ideas of the field appeared in the early work [8] and a very good introduction to the subject is [10], where the fundamental results of the theory are displayed alongside the main lines of research. One of these research lines seeks to determine, when X is compact and φ is continuous, what are the typical support of the maximizing measures (note that the existence of at least one maximizing measure is assured in this case by the compactness of the set of invariant probability measures in the weak-* topology). This is inspired by the classical conjecture of Mañé that, generically, the measures that minimize the Lagrangian action in Lagrangian flows are supported in periodic orbits.
There are some different conceptual approaches to this question. First, one may be interested in a specific dynamical property (as, for instance, Lyapunov exponents or rotation numbers) and so the potential is determined by the choice of the dynamical system. Examples of this are [11,9,7,6]. Another approach, followed for instance in [5,12,13], involves fixing a dynamical system f , usually with some specific dynamical condition like hiperbolicity or expansiveness, and varying the potential in a suitable space.
In this work we follow yet a different line, searching to understand how the maximizing measures behave when the potential is fixed, but the dynamics are allowed to change in a given space. In [3] it is shown that, if M is a compact Riemannian manifold of dimension n ≥ 2, then for any continuous φ : M → R there exists a dense set of homeomorphisms of M with a maximizing measure supported on aperiodic orbit, but in [1] it is shown that this set is meager. And in [2] it is shown that for a dense set of endomorphisms of the circle, there exists a φ maximizing measure supported on a periodic orbit. In this note we extend this last result, showing that Theorem 1. Let M be a compact Riemannian manifold and φ 0 : M → R continuous. Then there exists a dense subset A of End(M ) such that, for every f ∈ A there exists a φ 0 maximizing measure supported on a periodic orbit.
Where End(M ) is the set of continuous surjections of M endowed with the C 0 metric, d(f, g) = sup x∈M d(f (x), g(x)).
The strategy of the proof, similar to the one used in [2], is to make a series of local perturbations in order to obtain a periodic source with large φ 0 average while controlling the Birkhoff averages of the return map to the perturbation support. The proof of [2] relied on the local ordered structure of the domain, particularly in the definition of the support of the perturbations and in controlling the Birkhoff averages, two key points that were not adaptable to higher dimensions. In here we dealt with these difficulties by supporting perturbations in convex sets and analyzing the maximal Birkhoff sums on homothetic copies of the perturbation support, and by controlling the radial rate of escape from the periodic source.
The paper is organized as follows: In the next section we present some preliminary lemmas and notations, and in section 3 prove the theorem. Since the argument is perturbative, for a given endomorphism we analyze several possibilities, each dealt with in a different subsection, and show for each possibility how to construct a perturbed endomorphism close to f with the desired property.

Preliminaries
We start with some notations and preliminary results. Let M be a compact Riemannian manifold and End(M ) the set of endomorphisms of M , its continuous surjections. We endow End(M ) with its usual topology of uniform convergence and define the metric d(f, g) = sup x∈M (d(f (x), g(x)), f, g ∈ End(M ).
Given f ∈ End(M ) we denote by M inv (f ) the set of f invariant Borel probability measures, which is non-empty, convex and also compact in the weak-* topology. The subset of ergodic measures of M inv (f ) is denoted by M erg (f ).
Given φ : M → R continuous and f ∈ End(M ), we define P φ : M inv (f ) → R, P φ (µ) = φdµ. As the functional P φ is affine and M inv (f ) is a convex compact set, P φ must have a maximum point at an extremal point of M inv (f ). Since the extremal points of M inv are precisely the ergodic measures, there exists some The following lemma is a direct consequence of Atkinson's Lemma (see [4]) Lemma 2. Let φ : M → R be a continuous function, f ∈End(M ) and µ ∈ M erg (f ), such that, Then for µ-almost all x ∈ M , there exist n k → ∞ such that, We begin with the following simple result Proof. Let f 0 be an endomorphism, and let ε > 0. We will find some f with the stated property ε close to f 0 . First, let δ > 0 be such that, for all , where K is the ratio of the radii of the circunscribed and inscribed spheres in the n dimensional regular simplex. Since every n-dimensional differential manifold admits a triangulation and M is compact, we can assume that M has a triangulation 1 with finitely many triangles, such that each simplex has diameter less then δ and let 2 be a subtriangulation of 1 such that, for each ∆ i ∈ 1 there exists some∆ ji ∈ 2 which is contained in the interior of ∆ i .
Now we define f : M → M , in a way that f is a linear bijection in each triangle of 2 and such that, in local coordinates, f (∆ ji ) is a simplex that contains f 0 (∆ i ) and is contained in a sphere or radius ε/2. It should be immediate that f is a continuous surjection, since As f is linear in each simplex of 2 , the set {y ∈∆ j ∩ M : f (y) = x} is either empty or unitary, and therefore {y ∈ M : f 2 (y) = x} has cardinality smaller than or equal to the number of simplexes in 2 The structure of proof of Theorem 1 is the following. Let φ 0 : M → R be fixed. We start with an endomorphism f which we assume that, for every x ∈ M , the pre-image of x is finite and we construct successive small perturbations to produce an endomorphism f which is ε close to f and such that f has a φ 0 maximizing measure supported on a periodic orbit.
Let µ max ∈ M erg (f ) be a φ 0 maximizing measure and let φ = φ 0 − φdµ max , so that φdµ max = 0, and we remark that, for any endomorphism g, µ is a φ 0 maximizing measure if and only if it is a φ maximizing measure.
. If a 1 ≥ 0 we set y = x 1 , n = n k1 and we are done. If a 1 < 0, let n k2 > n k1 be such that f n k 2 (x 1 ) ∈ B ε (x) and such that S n k 2 f (x 1 ) > a 1 . Then, as S n k 2 f (x 1 ) = S n k 1 f (x 1 ) + S (n k 2 −n k 1 ) f (f n k 1 (x 1 )), we set y = f n k 1 (x 1 ) and n = n k2 − n k1 and we are done.
The next proposition is a consequence of the M f compactness.
Proof. This follows from lim sup 3. Contruction of the perturbed endomorphism Fix x ∈ supp(µ) and let ε > 0. There are two possibilities, 3.1. Case I. Let us show first how to construct f in the case I : Denote, for simplicity, B = B ε (x). We assume that for all y ∈ B and all n > 0, if f n (y) ∈ B then S n f (y) ≤ 0. From Lemma 4 there exists x 0 ∈ B and n 0 > 0 such that S n0 f (x 0 ) ≥ 0, and so S n0 f (x 0 ) = 0. Let 0 < n 1 ≤ n 0 be the first return of x 0 to B. Note that, as where the inequality comes from assuming that we are in case I, then S n1 f (x 0 ) ≥ 0 and, again from the assumption, S n1 f (x 0 ) = 0. Let T : M → M be a homeomorphism such that T (f n1 (x 0 )) = x 0 , and such that T is the identity outside of B, let f = T • f . Note that x 0 is a n 1 periodic point for f . Let µ 1 be the measure uniformly distributed on the points of the f orbit of x 0 .
On the other hand, if the return times of z to B are 0 ≤ t 0 < t 1 < t 2 ..... with t k → ∞, then so that lim n→∞ 1 n S n f (z) ≤ φdµ 1 for all z ∈ M and we have the result 3.2. Case II. Assume now we are in case II, and let a 0 = 1 n0 S n0 f (x 0 ) > 0. Denote by B ε [z] the closed ball with center z and radius ε. Let m 0 = m 0 (a 0 ) > n 0 > 0 be the integer from Proposition (1), and for each k ∈ {1, 2, . . . , m 0 } consider the compact sets For each k, let c k = sup z∈K k 1 k S k f (z) and let c = sup{c 1 , ..., c m0 }. Note that, by the choice of x 0 , c n0 ≥ a 0 . Furthermore, by Proposition (1), if n > m 0 then for all z ∈ M, 1 n S n f (z) ≤ a0 2 . This, and the choice of c implies that, for each z ∈ B ε [x 0 ] and n > 0 such that f n (z) ∈ B ε [x 0 ], we have 1 n S n f (z) ≤ c. We consider 2 distinct possibilities: The next lemma show us that the f invariant measure supported on the periodic orbit of q is a φ-maximizing measure.
Let z ∈ M and first assume that z is such that there exists some n such that where the second equality follows from the fact that f (y) = f (y) whenever y / ∈ B ε [x 0 ], and the inequality follows since the maximal φ average for f is 0, and from sup z∈M lim sup n→∞ 1 n S n f ( f n (z)) ≤ sup µ∈Minv(f ) φdµ. As 1 nq S nq f (q) > 0, we are done in this case. Now assume that there exists an increasing sequence of times N k → ∞, k ≥ 1 such that f i (z) belongs to B ε [x 0 ] if and only if i = N k for some integer k. Then it holds that 1 Where the inequality (1) follows from S k f (z) = S k f (z), as The previous lemma shows that, if z is a typical point of an f ergodic invariant measure µ, then lim n 1 n S n f (z) = φdµ ≤ 1 nq S nq f (q) and we are done.

3.2.2.
Case (b). There exists some z 1 ∈ B ε [x 0 ] and n z1 > 0 such that f nz 1 (z 1 ) ∈ ∂B ε [x 0 ], and such that 1 Let us call q 0 = f nz 1 (z 1 ). Since each point in M has finitely many preimages, the set P = ( Let q ∈P be a point which is closest to q 0 and let n q be such that f nq (q) = q 0 and Proof. By the choice of E,P ∩ E = {q} and so for any z = q in P ∩ E and n z such that f nz (z) = q 0 , 1 nz S nz f (z) is strictly smaller than c. Thus, by the continuity of f and φ, there exist δ 1 (z) > 0 such that if d(z, y) < δ 1 (z) we have nq S nq f (q). Moreover, for each δ 1 (z) there exist δ 2 (z) such that the connected component of f −i (B δ2(z) (q 0 )) which contain z is contained in B δ1(z) (z). Finally there exists δ 3 > 0 such that, if f −i (B δ3 (q 0 )) intersects E then there is some point of P in this component. By taking δ = min x∈P {δ 2 (x), δ 3 } we are done.
Denote the set E ∪ B δ [q 0 ] by I, we will construct a new endomorphism f = T • f , where T | M \I (z) = z , and such that there exist a f -periodic point in I whose average is strictly positive. Let Over D we define the following functions: By the Proposition 2, let W 0 be the connected component of f −1 2 (B δ [q 0 ]) which contains q, if z ∈ D and ψ(z) > ψ(q), then z ∈ W 0 . Denote by z max the point in W 0 that maximizes ψ(z). Choose α ∈ W 0 sufficiently close to z max such that the inequality is true, and such that f 2 (α) ∈ int(B δ [q 0 ]). Now we consider L to be the line segment joining α and f 2 (α), T 1 : M → M an homeomorphism mapping f 2 (α) to α, that is, T 1 (f 2 (α)) = α and such that T 1 is the identity outside V (L), where and δ 3 > 0, chosen such that V (L) is contained in the interior of I.
In figure (1), the shadow part is the neighborhood of the line segment L. We define now f by the composition f = T 1 • f . Note that α is a n q periodic point for f and that the φ average over the orbit of α is ψ(α) ≥ 1 nq S nq f (q) > 0. Yet the dynamics defined by f may have some new invariant measures whose φ average is strictly larger than ψ(α). Still, it should be clear that, as in the proof of Lemma 6 if z is such that the f orbit of z returns to D finitely many times, then lim sup n→∞ 1 n S n f (z) ≤ 0, and if z ∈ D returns infinitely-many times by f to the set I, but its orbit does not intersect W 0 (or just intersects it finitely many times), then if n 1 , n 2 , . . . are the return times to D, we have, by 2: and we remark that B δ [q 0 ] is disjoint from W 0 . So, if there is some future time n 1 > n q such that f n1 (z) ∈ W 0 , we can write n 1 = n q + k with f nq (z) = f 2 (z) ∈ B δ [q 0 ] and f nq+k (z) ∈ W 0 . The following estimate will be useful where the last inequality follows from (2).
3.3. The last pertubation. In order to finish the demonstration of the Theorem (1) we need to control the averages of those elements which have infinitely many returns on W α . In this section we construct a new pertubation T 2 such that α will be a source for the new endomorphism T 2 • f , and W α is contained in its basin of repulsion.
be the set of points who return to I by the function f . Over this set we define the following functions: The following propositions are immediate from the definitions: Let ψ max : I → R be the following function: ψ(y).
As the φ integral over the measure equidistributed over thef orbit of α is ψ(α), the final step in the proof of Theorem 1 is Proof. First, note that, if thef orbit of z does visit W 0 infinitely many times, then 3 and the same argument applied in lemmas 5 and 6 show the result.
Second, if thef orbit of z visits W 0 infinitely many times, but only visits W α a finite number of times, then using (4) and again using the reasoning in lemmas 5 and 6 we have the result. Now assume z ∈ W α returns to W α infinitely many times, and let n j the sequence of times such thatf nj (z) = (T 2 • f ) nj (x) ∈ I, where n 0 = 0 and n i+1 = n i + N ret (f ni (x)). Consider the following subsequences of (n k ) k∈N : • a j , where a 1 = 0 and a i+1 is the smallest integer larger than a i such that f na i+1 (x) is in W α , butf na i+1 −1 (x) is not.
• b j , where b i is the smallest integer larger than a i such thatf n b i −1 (x) is in W α , butf n b i (x) is not.
concluding the proof of the proposition and the theorem