E H El

Professor François Parreau

ON THE MAHLER MEASURE OF THE SPECTRUM OF RANK ONE MAPS ⋆

Keywords: Mathematics Subject Classification. Primary 37A05, 37A30, 37A40; Secondary 42A05, 42A55 Mahler measure, rank one maps, generalized Riesz products, affinity, Mahler Measure, Kakutani Dichotomy Theorem, Zygmund Dichotomy Theorem, flat polynomials, singular spectrum, simple Lebesgue spectrum, Hardy spaces, outer functions, inner functions

On the Mahler measure of the spectrum of rank one maps

Introduction

The purpose of this paper is to investigate the properties of the Mahler measure of the maximal spectral type of rank one maps and to discuss the question on whether the Mahler measure of the of the spectrum of rank one maps acting on a finite measure space is zero. According to our first main result, this is question concern the problem of the singularity of the spectrum of rank one maps acting on finite measure space [28, pp.14], [START_REF] Thouvenot | [END_REF]. Rank one maps have simple spectrum and using a random procedure, D. S. Ornstein produced a family of mixing rank one maps [START_REF] Ornstein | On the root problem in ergodic theory[END_REF]. It follows that Ornstein's class of maps may possibly contain a candidate for Banach-Rhoklin's problem whether there exists a map T acting on the probability space (Ω, A, P, ) with simple Lebesgue spectrum * . Unfortunately, in 1993, J. Bourgain proved that almost surely Ornstein's maps have singular spectrum [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF]. Subsequently, using the same methods, I. Klemes [START_REF] Klemes | The spectral type of staircase transformations[END_REF] showed that the subclass of staircase maps has singular maximal spectral type. In particular, this subclass contains the mixing staircase maps of Adams-Smorodinsky [START_REF] Adams | On Smorodinsky conjecture[END_REF]. Using a refinement of Peyrière criterium [START_REF] Peyrière | Étude de quelques propriétés des produits de Riesz[END_REF], I. Klemes & K. Reinhold proved that the rank one maps have a singular spectrum if the inverse of the cutting parameter is not in ℓ 2 (that is,

+∞ k=1 1 m 2 k
= +∞, where (m k ) ⊂ 2, 3, 4, • • • is the cutting parameter) [START_REF] Klemes | Rank one transformations with singular spectre type[END_REF].

This class contains the mixing staircase maps of Adams & Friedman [START_REF] Adams | Staircase mixing[END_REF]. In 1996, H. Dooley and S. Eigen adapted the Brown-Moran methods [30, pp.203-209] and proved that the spectrum of a subclass of Ornstein maps is almost surely singular [START_REF] Dooley | A family of generalized Riesz products[END_REF].

Later, el Abdalaoui-Parreau and Prikhod'ko extended Bourgain theorem [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF] by proving that for any family of probability measures in Ornstein type constructions, the corresponding maps have almost surely a singular spectrum [START_REF] El Abdalaoui | A new class of Ornstein transformations with singular spectrum[END_REF]. They obtained the same result for Rudolph's construction [START_REF] Rudolph | An Example of a measure-preserving map with minimal self-joining and applications[END_REF]. In 2007, el Abdalaoui showed that the spectrum of the rank one map is singular provided that the spacers (a j ) m k j=1 ⊂ N, are lacunary for all k [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF]. The author used the Salem-Zygmund Central Limit Theorem methods. As a consequence, the author presented a simple proof of Bourgain theorem [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF].

Five years later, by appealing to the martingale approximation technique, C. Aistleitner and M. Hofer [START_REF] Aistleitner | On the maximal spectral type of a class of rank one transformations[END_REF] proved a counterpart of the result in [START_REF] El Abdalaoui | A new class of rank-one transformations with singular spectrum[END_REF]. Precisely, they proved that the spectrum of the rank one maps is singular provided that the cutting parameter (m k ) ∈ N * and the spacers (a j ) m k j=1 ⊂ N satisfy: i) log(m kn ) h kn converge to 0;

ii) the proportion of equal terms in the spacers is at least c.m kn for some fixed constant c and some subsequence (k n ).

Seven years ago [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF], e. H. el Abdalaoui and M. Nadkarni produced an infinite product formula for the Mahler measure of the absolutely continuous part of the generalized Riesz product related to the Mahler measure of the polynomials (P k ) k∈N associated to it. BY appealing to the H p theory, they proved that this infinite product is exactly the Mahler measure of its . They further established that the infinite product of (P k ) k∈N converge in L 1 to the square root of the absolutely continuous part of the generalized Riesz product. From this, they conjectured that the logarithm of the absolutely continuous part of the spectrum of any rank one map acting on a space of finite measure is not integrable. Moreover, they proved that the Banach's problem, Littlewood problem and Mahler problem are equivalent in the class of rank one maps acting on a space of infinite measure.

Banach's problem. Following Ulam [59, p.76], Banach asked:

Question 1 (Banach Problem). Does there exist a square integrable function f (x) and a measure preserving transformation T (x), -∞ < x < ∞, such that the sequence of functions {f (T n (x)); n = 1, 2, 3, • • • } forms a complete orthogonal set in Hilbert space? †

Obviously, as pointed by Rokhlin, the map n ∈ Z → n+1 with the indicator function of {1} answer the question. But the map is dissipative. For the conservative case, el Abdalaoui obtained also an affirmative answer by producing an infinite rank one map [START_REF] El Abdalaoui | Ergodic Banach problem, flat polynomials and Mahler's measures with combinatorics[END_REF]. The proof is based on the ideas from [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF] [6] combined with the singer's theoretical number theory theorem and tools from extremal harmonic analysis.

The strategy is accomplished, as in [START_REF] El Abdalaoui | A non-singular transformation whose Spectrum has Lebesgue component of multiplicity one[END_REF], by producing a sequence of flat analytic polynomials with coefficients 0, 1. But the polynomials are only flat in L 1 -sense instead in [START_REF] El Abdalaoui | A non-singular transformation whose Spectrum has Lebesgue component of multiplicity one[END_REF], the polynomials are ultra-flats with positive coefficients.

The most famous Banach problem in ergodic theory (which should be attributed to Banach and Rokhlin) asks if there is a measure preserving transformation on a probability space which has simple Lebesgue spectrum. A similar problem is mentioned by Rokhlin in [54, p.219]. Precisely, Rokhlin asked on the existence of an ergodic measure preserving transformation on a finite measure space whose spectrum is Lebesgue type with finite multiplicity. Later, Kirillov in his 1966's paper [START_REF] Kirillov | Dynamical systems, factors and group representations, (Russian) Uspehi Mat. Nauk[END_REF] wrote "there are grounds for thinking that such examples do not exist". However he has described a measure preserving action (due to M. Novodvorskii) of the group ( ∞ j=1 Z) × {-1, 1} on the compact dual of discrete rationals whose unitary group has Haar spectrum of multiplicity 2. Similar group actions with higher finite even multiplicities are also given.

Subsequently, finite measure preserving transformation having Lebesgue component of finite even multiplicity have been constructed by J. Mathew and M. G. Nadkarni [START_REF] Mathew | A measure-preserving transformation whose spectrum has Lebesgue component of multiplicity two[END_REF], Kamae [START_REF] Kamae | /77[END_REF], M. Queffelec [START_REF] Queffélec | Substitution dynamical systems-spectral analysis[END_REF], and O. Ageev [START_REF] Ageev | Dynamical System With an Even-Multiplicity Lebesgue Component in the Spectrum[END_REF]. Fifteen years later, M. Guenais [START_REF] Guenais | Morse cocycles and simple Lebesgue spectrum Ergodic Theory Dynam[END_REF] used a L 1 -flat generalized Fekete polynomials on some torsion groups to construct a group action with simple Lebesgue component. A straightforward application of Gauss formula yields that the generalized Fekete polynomials constructed by Guenais are ultraflat. Very recently, el Abdalaoui and Nadkarni strengthened Guenais's result [START_REF] El Abdalaoui | A non-singular transformation whose Spectrum has Lebesgue component of multiplicity one[END_REF] by proving that there exist an ergodic non-singular dynamical system with simple Lebesgue component. However, despite all these efforts, it is seems that the question of Rokhlin still open since the maps constructed does not have a pure Lebesgue spectrum.

Here, we are able to extend partially Kakutani-Zygmund dichotomy theorem by proving that the maximal spectral type of the rank one map is either singular or its absolute continuous part is equivalent to the Lebesgue measure, depending † Professor M. Nadkarni pointed to me that the question contain an oversight. The sequence of functions should be bilateral, that is, n ∈ Z.

whether the product of Mahler measure of (P k ) k∈N converges or diverges (Theorem 3.2 below). This combined with the recent result of el Abdalaoui and Nadkarni in [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF] and [START_REF] El Abdalaoui | Notes on the flats polynomials[END_REF] allows us to establish that the rank one maps has a singular spectrum provided that each P k has less than c.h k-1 zeros bigger than 1 in absolute value where c is a positive constant less than one.

Furthermore, we are able to give a negative answer to Nadkarni question by proving that the integral of the absolute part of any non trivial generalized Riesz products is strictly less than 1. Furthermore, using a deep result of Bourgain [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF], we establish that there is a new subclass of rank one maps acting on finite and infinite measure space for which the Mahler measure is zero. For this subclass the cutting parameter satisfies m k = θ(k β ), for some β ≤ 1. As a consequence, we obtain that the spectrum of any map in this class is singular. However, we are not able to answer the question on whether the Mahler measure on any rank one acting on probability space is zero. Nevertheless, we made the following conjecture.

Conjecture 1. Let (X, B, P, T ) be a rank one map and assume that P(X) < +∞. Then, the Mahler measure of its spectrum is zero.

Obviously, this conjecture is related to Klemes-Reinhold's conjecture [START_REF] Klemes | Rank one transformations with singular spectre type[END_REF] on the spectrum of rank one. Indeed, therein the authors conjectured that all rank one maps have singular spectrum and, in the same spirit, C. Aistleitner and M. Hofer wrote in the end of their paper "several authors believe that all rank one transformations could have singular maximal spectral type." [START_REF] Aistleitner | On the maximal spectral type of a class of rank one transformations[END_REF]. It seems that this conjecture was formulated since Baxter result [START_REF] Baxter | On the class of ergodic transformations[END_REF], [START_REF] Thouvenot | [END_REF]. Of-course, the conjecture stand only for rank one acting on finite measure space.

The paper is organized as follows. In section 2 we recall the definition of rank one map and some standard facts on the notion of Mahler measure. This allows us to state our main results in section 3 . In section 4, we review some basic facts on the affinity and the mutual singularity of two probability measures. Finally, in section 5, we present the proofs our ours main results.

rank one maps by cutting and stacking methods

Using the cutting and stacking method described in [START_REF] Friedman | Replication and stacking in ergodic theory[END_REF], [START_REF] Friedman | Introduction to Ergodic Theory[END_REF], one can construct inductively a family of measure preserving maps, called rank one maps, as follows Let B 0 be the unit interval equipped with Lebesgue measure. At stage one we divide B 0 into m 0 equal parts, add spacers and form a stack of height h 1 in the usual fashion. At the k th stage we divide the stack obtained at the (k -1) th stage into m k-1 equal columns, add spacers and obtain a new stack of height h k . If during the k th stage of our construction the number of spacers put above the j th column of the (k -1) th stack is a

(k-1) j , 0 ≤ a (k-1) j < ∞, 1 ≤ j ≤ m k , then we have h k = m k-1 h k-1 + m k-1 j=1 a (k-1) j .
Proceeding in this way we get a rank one map T on a certain measure space

pk towers Bk • • • • • • . . . . . . . . . . . . Bk+1 a (k) 2 a (k) 1 a (k) i a (k) pk • • • • • • . . . • • • Fig. (k + 1)th tower.
Stade k :

(X, B, | . |) which may be finite or σ-finite depending on the number of spacers added.

The construction of a rank one map thus needs two parameters, (m k ) ∞ k=0 (cutting parameter), and ((a

(k) j ) m k j=1 ) ∞ k=0 (spacers parameter). Put T def = T (m k ,(a (k) j ) m k j=1 ) ∞ k=0
In [START_REF] Choksi | The maximal spectral type of rank one transformation[END_REF] and [START_REF] Klemes | Rank one transformations with singular spectre type[END_REF] it is proved that the spectral type of this map is given (up to possibly some discrete measure) by

dµ = W * lim n k=1 P k 2 dλ, (1)
where

P k (z) = 1 √ m k   1 + m k -1 j=1 z -(jh k + j i=1 a (k) i )   ,
λ denotes the normalized Lebesgue measure on the circle group T and W * lim denotes weak * limit in the space of bounded Borel measures on T.

As mentioned by Nadkarni in [START_REF] Nadkarni | Spectral Theory of Dynamical Systems[END_REF], the infinite product

+∞ l=1 P j l z)| 2
taken over a subsequence j 1 < j 2 < j 3 < • • • , also represents the maximal spectral type (up to discrete measure) of some rank one maps. In case j l = l for infinitely many l, the maps acts on an infinite measure space.

We will denote by ξ n (x)

def = 1 √ |Bn| 1 Bn (x)
the characteristic function of the n thbase, normalized so that the 2-norm equals 1. We recall that the associated Koopman operator U T ξ is define by U T ξ(x) = ξ(T -1 x) on L 2 (X) and, for any ξ ∈ L 2 (X) there corresponds a positive measure µ ξ on T, the unit circle, defined by

µ ξ (n) = U n T ξ, ξ . With this notation, put µ n = µ ξn . Notice that C = {{T k (B n )} hn-1 k=0 } ∞ n=0 (1.2)
generates a dense subalgebra of the Borel σ-algebra, (here we are using the metric (modulo sets of measure zero) given by d(A, B) = A△B ). Then the subspace generated by the span of

{U k T (ξ n ) : 1 ≤ n < ∞, 0 ≤ k < h n } = span of {1 T k (Bn) : 1 ≤ n < ∞, 0 ≤ k < h n } is dense in L 2 (X).
We end this section by introducing the notion of Mahler measure and stating our second main result. The Mahler measure of P k is defined by

M (P k ) = exp ˆT log P k (z) dz .
Using Jensen's formula [START_REF] Rudin | Real and complex analysis[END_REF], it can be shown that

M (P k ) = 1 √ m k |α|>1 |α|,
where, α denoted the zero of the polynomial √ m k P k . In this definition, an empty product is assumed to be 1 so the Mahler measure of the non-zero constant polynomial P (x) = a is |a|. A nice account on the subject may be founded in [24, pp.2-11], [START_REF] Borwein | Computational Excursions in Analysis and Number Theory[END_REF].

Here are some elementary properties of the Mahler measure. But, we provide a proof for the reader's convenience.

Proposition 2.1. Let (X, B, ρ) be a probability space. Then, for any two positive functions f, g ∈ L 1 (X, ρ), we have i) M (f ) is a limit of the norms ||f || δ as δ goes to 0, that is,

||f || δ def = ˆfδ dρ 1 δ ---→ δ→0 M (f ), provided that log(f ) is integrable. ii) If ρ f > 0 < 1 then M (f ) = 0. iii) If 0 < p < q < 1, then f p ≤ f q . iv) If 0 < p < 1, then M (f ) ≤ f p . v) lim δ-→0 ˆf δ dρ = ρ f > 0 . vi) M (f ) ≤ f 1 . vii) M (f g) = M (f )M (g).
Proof. We start by proving ii). Without loss of generality, assume that f ≥ 0 and put

B = f > 0 , and let δ = 1/k be in ]0, 1[, k ∈ N * . Then 1/(1/δ) + 1/(1 -δ) = 1/k + (k -1)/k = 1.
Hence, by Hölder inequality, we have

ˆfδ dρ = ˆf1/k .1 B dρ (2) ≤ ˆ(f 1/k ) k dz 1/k ˆ1k/k-1 B dz k-1/k (3) ≤ ˆfdρ 1/k ˆ1B dz k-1/k (4)
≤ ˆfdρ

1/k ρ(B) (k-1)/k
Therefore we have proved

||f || δ ≤ ˆfdρ ρ(B) (1-δ)/δ (5) ≤ ˆfdρ ρ(B) k-1 -----→ k→+∞ 0,
To prove i), apply the Mean Value Theorem to the following functions

δ -→ x δ , if x ∈]0, 1[; t -→ t δ , if x > 1,
Hence, for any δ ∈]0, 1[ and for any x > 0, we have

x δ -1 δ ≤ x + log(x) .
Furthermore, it is easy to see that

f δ -1 δ = e δ log(f ) -1 δ ---→ δ→0 log(f ),
and, by Lebesgue Dominated Convergence Theorem, we get that

ˆfδ -1 δ dρ ---→ δ→0 ˆlog(f)dρ.
On the other hand, for any δ ∈]0, 1[

f δ = exp 1 δ log ˆfδ dρ ,
and for a sufficiently small δ, we can write

1 δ log ˆfδ dρ ∼ ˆfδ -1 δ dρ since log(x) ∼ x -1 as x -→ 1. Summarizing we have proved lim δ-→0 ||f || δ = exp ˆlog(f)dρ = M (f ).
For the proof of iii) and iv), notice that the function x → x q p is a convex function and x → log(x) is a concave function. Applying Jensen's inequality to

ˆ f p dρ we get f p ≤ f q , ˆlog( f )dρ ≤ log f p ,
and this finishes the proof, the rest of the proof is left to the reader.

Szegö-Kolmogorov-Krein theorem established a connection between a given measure and the Mahler measure of its derivative. Precisely, we have ). Let σ be a finite positive Baire measure on the unit circle and let h be the derivative of σ with respect to normalized Lebesgue measure. Then, for any r > 0,

M (h) = inf P 1 -P r r = inf P ˆ 1 -P r h(z)dz ,
where P ranges over all analytic trigonometric polynomials with zero constant term. The right side is 0 if log(h) is not integrable.

Clearly, Szegö-Kolmogorov-Krein theorem gives an alternative definition to Mahler measure (that is, the Malher measure of a given measure is the Mahler measure of its derivative). For other definitions, we refer the reader to [START_REF] Dégot | Finite-dimensional Mahler measure of a polynomial and Szegö's theorem[END_REF].

The Mahler measure is very useful in number theory and the use of this quantity in number theory is essentially due to Mahler [START_REF] Mahler | An application of Jensen's formula to polynomials[END_REF].

Following Helson and Szegö [START_REF] Helson | A problem in prediction theory[END_REF], Szegö-Kolmogorov-Krein theorem solved the first problem of the theory of prediction and the second problem of this theory was solved by Kolmogorov as follows ). Let σ be a finite positive Baire measure on the unit circle and let h be the derivative of σ with respect to normalized Lebesgue measure. Then,

M 0 (h) = inf P ∈M0 1 -P 2 2 = ˆ1 h dz -1
, where M 0 is a closed subspace generated by {z n , n ∈ Z \ {0}}. The right side is 0 if log(h) is not integrable.

In his 1984's paper [START_REF] Nakazi | Two problems in prediction theory[END_REF], Nakazi extended simultaneously Kolmogorov prediction theorem and the following result due to Nakazi and Takahashi [START_REF] Nakazi | Prediction n units of time ahead[END_REF] Theorem 2.4 (Nakazi-Takahashi [START_REF] Nakazi | Prediction n units of time ahead[END_REF], [START_REF] Pourahmadi | A prediction problem in L 2 (w)[END_REF].). Let σ be a finite positive Baire measure on the unit circle and let h be the derivative of σ with respect to normalized Lebesgue measure. Then,

M n (h) = inf P ∈Mn 1 -P 2 = n k=0 |α k | 2 1 2 ,
where M n is a closed subspace generated by {z k , k ≥ (n + 1)} and (α k ) +∞ k=0 is the Fourier coefficients of the associated outer function φ to h (that is,

h = |φ| 2 = +∞ k=0 α k z k | 2 ). The right side is 0 if log(h) is not integrable.
Nakazi Theorem generated considerable interest in computing the predicator error when the index set is {1, 2, • • • } with finitely many points of Z added or deleted. For a recent results we refer the reader to [START_REF] Pourahmadi | A prediction problem in L 2 (w)[END_REF], [START_REF] Cheng | Some extremal problems in L p (w)[END_REF] and the references therein.

Let us mention that the fundamental ingredients in the proofs of the previews results are based on the Hardy space Theory. For 0 < p < ∞, the Hardy space H p is the L p (dz)-closure of {1} + P 0 , where P 0 is the manifold of trigonometric polynomials whose frequencies are in 1, 2, 3, • • • , . H ∞ is defined to be the weak star closure of {1} + P 0 in L ∞ (dz).

We further have the following Zygmund's theorem [40, p.96]

Theorem 2.5 (LlogL Zygmund's theorem.). Let |f (θ)| log + (|f (θ)) ∈ L 1 (-π, π). Then F (z) = ˆeit + z e it -z f (t)dt belonging to H 1 .
Based on Szegö-Kolmogorov-Krein theorem (Theorem 2.2) and its allies, el Abdalaoui-Nadkarni proved the following theorem [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF] that we need here.

Theorem 2.6. Let T be a rank one maps with the cutting parameter (m k ) and spacers parameter ((a

(k) j ) m k j=1
) ∞ k=0 and let µ = j≥0 P j 2 be its spectral type. Then

M dµ dz = +∞ j=0 M P 2 j = lim n→∞ exp ˆS1 log n j=1 |P j (z)| 2 dz ,
where dµ dz is a derivative of µ.

We also need the following theorem from [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF].

Theorem 2.7. Let µ be a generalized Riesz product based on the sequences of polynomials P k , that is,

µ = +∞ k=0 |P k (z)| 2 def = W * lim N k=0 |P k (z)| 2 dz,
where

P k (z) = 1 √ m k m k -1 j=0 z jh k +s(k,j) , s(k, 0) = 0 and s(k, j) = j i=1 a (k) i .
Then the product

Q N = N k=1 |P k (z)| converge in L 1 (dz) to dµ dz .
We shall also need the following Proposition [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF]. Given two subsequences N 1 and N 2 we construct three generalized Riesz product as follows

µ 1 = k∈N1, |P k | 2 , µ 2 = k∈N2, |P k | 2 and ν = k∈N1∪N2 |P k | 2 .
We shall establish some relation between the absolutely continuous part of all those three measures. Indeed, we have Lemma 2.9. dν/dz = dµ 1 /dz dµ 2 /dz.

Proof. For any n ∈ N, put

Q n = k∈N 1 k≤n |P k | and R n = k∈N 2 k≤n |P k |.
Then, by Theorem 2.7, Q n R n converge to dµ/dz almost everywhere over a subsequence and the same holds for the subproducts Q n and R n . Thus, over a common subsequence for all three cases the convergence holds almost everywhere. Whence dν/dz = dµ 1 /dz dµ 2 /dz, which proves the lemma.

We have also the following fundamental lemma.

Lemma 2.10. There is a subsequence

1 Q ni n≥0 and ξ in L 2 (µ) such that 1 Q ni n≥0 converge weakly to ξ in L 2 (µ).
Proof. Observe that we have

µ = Q 2 n dµ n , with µ n = +∞ k=n |P k (z)| 2 . ( 6
)
This combined with Cauchy-Schwarz inequality yields

ˆ1 Q n dµ ≤ ˆ1 Q 2 n dµ 1 2 = ˆdµ n 1 2 = 1.
Hence, the sequence 1

Q n n≥0
is bounded in L 2 (dµ) and this implies that there is a subsequence that converges weakly to some function ξ in L 2 (µ). The proof of the lemma is complete.

Main results

We start by stating our first main result.

Theorem 3.1 (First main result). Let (X, B, P, T ) be a rank one map. Then, the spectral type is either singular or its absolute continuous part is Lebesgue.

For our second main result, by appealing to Theorem 2.6, we are able to extended Kakutani theorem and to prove the analogous of Zygmund dichotomy theorem. Precisely, we have Theorem 3.2 (Second main result). The spectral type of any rank one maps is either singular or its absolute continuous part is equivalent to Lebesgue measure with positive Mahler measure, that is, λ ∼ dµ dλ (θ)dλ or µ ⊥ λ according as j≥0 M (P j ) converge or diverge.

Our third main result gives a negative answer to the question raised by M. Nadkarni. We state it as follows Notice that the spectral measure of any indicator function in any dynamical system with phase space of finite measure has a point mass at 1. Therefore, for any rank one map acting on finite space its spectral type satisfy

ˆdν dz dz ≤ 1 - 1 |X| ,
since µ has a mass 1/|X| at 1.

The proof of Theorem combined with the deep result of Bourgain [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF], allows to obtain the following.

Theorem 3.4 (fourth main result ). For any rank one maps with the cutting parameter (m j ) satisfying m j = θ(j β ), for some β ≤ 1, we have

+∞ j=0 M P j = 0.
Let us point out that our methods combined with Bourgain proposition [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF], allows us to prove:

Corollary 3.5. Any rank one maps with the cutting parameter (m j ) satisfying m j = θ(j β ), for some β ≤ 1 have a singular spectrum.

We recall that in the standard asymptotic notation called Bachmann-Landau notation [START_REF] Flajolet | Analytic Combinatorics[END_REF]: m j = θ(j β ) means that there exits c, C > 0 and a large j 0 such that for any j ≥ j 0 , we have c ≤ m j j β ≤ C.

Furthermore, let us mention that el Abdalaoui and Nadkarni proved As a consequence, we have Corollary 3.7. If each P k has less than c.h k-1 zeros bigger than 1 in absolute value where c is a positive constant less than one, then the associated rank one map has a singular spectrum.

Remark 3.8. Almost all know results on the singularity of rank one maps can be derived from our method combined with our first main result.

Affinity between two measures

The affinity or Hellinger integral between two finite measures is defined by the integral of the corresponding geometric mean. It was introduced and studied in a series of papers by K. Matusita [START_REF] Matusita | Decision rules, based on the distance for problems of fit, two samples, and estimation[END_REF], [START_REF] Matusita | A distance and related statistics in multivariate analysis, Multivariate Analysis[END_REF], [START_REF] Matusita | On the notion of affinity of several distributions and some of its applications[END_REF] and it is also called Bahattacharyya coefficient [START_REF] Bhattacharyya | On a measure of divergence between two statistical populations defined by their probability distributions[END_REF]. The affinity between two probability measures σ and ρ is defined by

(7) G(σ, ρ) = ˆ dσ dτ . dρ dτ dτ.
This definition does not depend on τ . The affinity is related to the Hellinger distance as it can be defined as

H(σ, ρ) = 2(1 -G(σ, ρ)). Note that G(σ, ρ) satisfies (by Cauchy-Schwarz inequality) 0 ≤ G(σ, ρ) ≤ 1.
It is an easy exercise to see that G(σ, ρ) = 0 if and only if σ and ρ are mutually singular (denoted by σ⊥ρ.) and G(σ, ρ) = 1 holds if and only if σ and ρ are equivalent.

Using the affinity, T. Kamae in [START_REF] Kamae | /77[END_REF] and Coquet-Mandés-France-Kamae in [START_REF] Coquet | Sur la mesure spectrale de certaines suites arithmétiques[END_REF] introduced a tools to study the spectral proprieties of q-multiplicative sequences. They proved the following result.

Theorem 4.1 (Coquet-Mandès-France-Kamae [START_REF] Coquet | Sur la mesure spectrale de certaines suites arithmétiques[END_REF]). Let (σ n ) and (ρ n ) be two sequences of probability measures on the circle weakly converging to the probability measures σ and ρ respectively. Then [START_REF] Adams | On Smorodinsky conjecture[END_REF] lim sup

n-→+∞ G(σ n , ρ n ) ≤ G(σ, ρ),
Let us mention that the affinity methods can be used to establish the celebrated Kakutani theorem and Hajek-Feldmen theorem [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF].

G. Ritter [START_REF] Ritter | On Kakutani's dichotomy theorem for infinite products of not necessarily independent functions[END_REF] and Brown-Moran [START_REF] Brown | On orthogonality of Riesz products[END_REF] use the same methods in the context of the classical Riesz products. They mentioned that the dissociation can be viewed as a analogous of the stochastic independence. Thus, in this context, the analogous of Kakutani and Hajek-Feldmen theorems is known as Zygmund dichotomy theorem [60, pp.263-264] Later, using the affinity methods combined with the Bourgain tools, el Abdalaoui [START_REF] El Abdalaoui | La singularite mutuelle presque sûre du spectre des transformations d'Ornstein[END_REF] established that almost surely the spectral types of Ornstein maps are mutually singular.

Here, we are able to obtain a refinement of the Coquet-Mandés-France-Kamae theorem (Theorem 4.1) by proving that the sequence k≥0 |P k | converge in L 1 to the square root of the derivative of µ. It is turn out that this result is a strong ingredient in the proof of our main results. We shall need also the following lemma inspired by Bourgain and Kilmer-Seaki methods.

Lemma 4.2. Let ρ be a probability measure on the circle T and σ n = f n dρ be a sequence of probability measures on T such that (1) σ n converge weakly to some probability measure τ .

(2) f n is positive almost everywhere with respect to τ and ˆ1 f n dτ is a bounded sequence. Then

τ ⊥ ρ ⇐⇒ lim n-→+∞ ˆ f n dρ = 0.
Proof. Suppose τ and ρ are mutually singular. Then, the affinity G(τ, ρ) is zero, which, by Coquet-Mandés-France-Kamae theorem (Theorem 4.1), gives lim

n-→+∞ ˆ f n dρ = 0.
Conversely, let ε be a positive number, then there exists a large integer n 0 such that

ˆ f n dρ < ε.
But, by our assumption (2), there exists C > 0, such that,

ˆ1 f n dτ < C.
Without loss of generality, let us assume that C = 1. Therefore, by Cauchy-Schwarz inequality, we have

ˆ1 √ f n dτ < 1.
Hence,

ˆ f n dρ ˆ1 √ f n dτ < ε,
and this implies that τ and ρ are mutually singular, by the lemma below.

The lemma is due to Kilmer and Saeki [START_REF] Kilmer | On Riesz product measures; mutual absolute continuity and singularity[END_REF]. We include the proof for the reader's convenience.

Lemma 4.3 (Kilmer-Saeki [START_REF] Kilmer | On Riesz product measures; mutual absolute continuity and singularity[END_REF]). Let ρ and τ be a nonnegative finite measures on measurable space X. Then the following properties are equivalent:

a) ρ ⊥ τ.
b) Given ε > 0, there exists nonnegative measurable function f on X such that f > 0, τ -a.e. and such that ˆfdρ ˆ1 f dτ < ε.

Proof. Suppose a) obtains. Then, there exists two disjoint measurable sets A, B such that ρ(A) = ρ(X) and τ (B) = τ (X). For a given ε > 0, put

f = ε1 A + 1 ε 1 B .
We thus get ˆfdρ ˆ1 f dτ = ερ(X)ετ (X) < ε, and this establish b).

Conversely suppose b) obtains. Let τ ′ be a large measure such that τ ′ ≤ ρ and τ ′ ≤ τ . Given ε > 0, let f be a function furnished by b). Since f > 0, τ -a.e., we also have f > 0, τ ′ -a.e.. Therefore, Cauchy-Schwarz inequality combined with b) yields

τ ′ (X) = ˆ f 1 √ f dτ ′ (9) ≤ ˆfdτ ′ 1 2 ˆ1 f dτ ′ 1 2 (10) ≤ ˆfdρ 1 2 ˆ1 f dτ 1 2 (11)
≤ √ ε.
Since ε > 0 was arbitrary, this gives that τ ′ = 0, which means that, ρ ⊥ τ. The proof of the lemma is complete.

Let us recall the following important and classical fact from Probability Theory connected to the notion of the uniform integrability.

Definition 4.4. Let (X, B, P) be a probability space and p ∈

[1, +∞[. A sequence {f n , n ≥ 1} in L p (X) is said to be L p (X) uniformly integrable if lim c-→+∞ sup n∈N ˆ |fn|>c f n p dP = 0. It is well-known that if sup n∈N f n p+ε < +∞, (12)
for some ε positive, then the sequence {f n } is L p uniformly integrable.

It is obvious that the almost everywhere convergence does not in general imply the convergence in L p (X). Nevertheless, it is well known that the condition of domination insure such convergence (Lebesgue's Dominated Convergence Theorem) but in the absence of domination the following Vital Convergence Theorem allows us to obtain the convergence in L p (X) provided that the sequence is uniformly integrability.

Theorem 4.5 (Vitali Convergence Theorem ). Let (X, B, P) be a probability space and {f n } be a uniformly integrable sequence in L p (X) which converges almost surely to some function f . Then f is in L p (X) and

f n -f p -----→ n→+∞ 0.
For the proof of Theorem 4.5 we refer the reader to [55, pp.134-135], [57, p. 165-167] .

Proof the main results

Let σ and τ be two measures on the circle. Then, by Lebesgue decomposition of σ with respect to τ , we have

σ = dσ dτ dτ + σ s ,
where σ s is singular to τ and dσ dτ is the Radon-Nikodym derivative. In the case of generalized Riesz product µ, we are able to establish that the following.

Proposition 5.1. Let N = k 0 < k 1 < k 2 < • • • be a subsequence and ν = k∈N |P k | 2 . Then, we have ˆ dν dz dz ≤ ˆ P k0 (z) dz 1 2 . (13)
In [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF], J. Bourgain proved the following proposition. Proposition 5.2 (Bourgain [START_REF] Bourgain | On the spectral type of Ornstein class one transformations[END_REF]). Let n be a positive integer and 0 ≤ l 1 < l 2 < • • • < l n be a finite sequence of non negative integers. Then, there exists an absolute constant c > 0 such that

ˆ 1 √ n n j=1 z li dz ≤ 1 -c log(n) n .
Proposition 5.1 combined with Proposition 5.2 yields the following

Corollary 5.3. Let N = k 0 < k 1 < k 2 < • • • be a subsequence and ν = k∈N |P k | 2 . Then, ˆ dν dz dz ≤ 1 -c log(m k0 ) m k0 1 2 , (14)
for some absolute constant c > 0. From Proposition 5.1 combined with Lemma 2.9, we have the following lemma.

Lemma 5.4. For any k ∈ N * , we have

ˆ2k dµ dz dz ≤ k-1 j=0 1 -c log(m j ) m j 1 2k . (17)
Proof. Let n 0 be a positive integer and N j , j = 1, • • • , n 0 be a partition of N. Denoted by µ j the generalized Riesz product construct over the subsequence N j for each j = 1, 2, • • • , n 0 . Hence, by Lemma 2.9, we deduce that dµ/dz is the product of dµ i /dz, j = 1, 2, • • • , n 0 . Indeed, one may take the Euclidian partition given by classifying the integers modulo k. Hence, by Lemma 2.9, we have

dµ dz = k i=1 dµ i dz .
Consequently, by Proposition 5.1 we get

ˆ dµ i dz dz 2 ≤ ˆ|P i |dz, for i = 0, 1, • • • , k -1. ( 18
)
From this and Hölder inequality, we conclude that,

ˆ2k dµ dz dz ≤ k-1 j=0 1 -c log(m j ) m j 1 2k . ( 19
)
which proved the lemma. Now, let us proved Theorem 3.4.

Proof of Theorem 3.4. It is easy to check from (??) that, for any k ≥ 1,

M dµ dz ≤ k-1 j=0 ||P j || 1 . (20)
This combined with Proposition 5.2, yields

M dµ dz = k-1 j=0 M dµ j dz ≤ k-1 j=0 1 -c log(m j ) m j .
Taking into account that m j = θ(j β ), with β ≤ 1, it follows that

M dµ dz ≤ k-1 j=0 1 -c log(m j ) m j ----→ j→+∞ 0,
which proves the theorem.

At this point let us present the proof of Theorem 3. .

Moreover, by Cauchy-Schwarz inequality, we can write

ˆ P m0 (z) dz < 1,
Since the equality in Cauchy-Schwarz inequality holds if and only if P m0 (z) is a constant polynomial which is impossible in our case. The proof of the theorem is complete.

Remark 5.5. The proof of Theorem 3.3 can be obtained also by combining Propositions 5.1 and 5.2.

We shall need the following lemma Lemma 5.6. Let τ be a finite measure on a Borel space X. Suppose that the sequences of positive functions (f n ) n≥0 and

1 f n in L 1 (τ ) converge in L 1 (τ ) to 1.
Then, there is a subsequence (f ni ) i≥0 such that the Mahler measure of f ni converge to 1, that is,

exp ˆlog f ni dτ ----→ i→+∞ 1.
Proof. By our assumption there is a subsequence (f ni ) i≥0 such that f ni and 1 f ni converge almost everywhere to 1. Therefore,

log + f ni ---→ i→∞ 0 and log -f ni ---→ i→∞ 0,
where log + (t) = max(0, log(t)) and log -(t) = max(0, -log(t)). We claim that the sequence log(f ni ) i≥0 converge to 0 in L 1 (λ). Indeed, the sequences log + f ni and log -f ni are uniformly integrable. Since, for any positive number C we have

log + f ni > C = log + f ni > C, f ni ≤ 1 log + f ni > C, f ni > 1 (21) = f ni > e C , f ni > 1 = f ni > e C .
Hence,

1 2 ˆ log + fn i >C log + f ni dτ = ˆ fn i >e C log f ni dτ (22) ≤ ˆ fn i >e C f ni dτ,
since log(x) ≤ x for any x ≥ 1. This combined with Cauchy-Schwarz inequality yields

1 2 ˆ log + fn i >C log + f ni dτ ≤ τ f ni > e C 1 2 ˆfni dτ 1 2
, and, by Markov inequality, we get

1 2 ˆ log + fn i >C log + f ni dτ ≤ e -C/2 sup i≥0 ||f ni || 1 (23) ≤ e -C/2 K -----→ C→+∞ 0,
where K is some positive constant such that sup

i≥0 ||f ni || 1 < K. It is remains to prove that log -f n0 is uniformly integrable. Notice that log -f ni > C = log -f ni > C, f ni < 1 log -f ni > C, f ni ≥ 1 = 1 f ni > e C , f ni < 1 = 1 f ni > e C . (24)
Therefore, in the same manner as before, we obtain

ˆ log -fn i >C log -f ni dτ ≤ e -C/2 K ′ -----→ C→+∞ 0,
where K ′ is some positive constant such that sup

M τ (f ni ) = exp ˆlog f ni dτ ---→ i→∞ 1,
and the proof of the lemma is achieved.

Proof of Theorem 3.2. Put e = j≥0 M (P j ) , Q n = n j=0 |P j |, f = dµ dz and B = f 2 > 0 .
Suppose e is positive. Then, by Proposition 2.1, it follows that

λ(B) = 1,
which means that the absolutely continuous part of µ is equivalent to Lebesgue measure λ. Conversely, assume that λ is equivalent to dµ dλ dλ then for any nonnegative integer n, λ is equivalent to dµn dλ dλ, and by Theorem 2.7, we can write

= +∞ j=n |P j |,
in the sense of L 1 (dz). Put

φ n = dµ n dλ and ν n = min(1, φ 2 n ) dz, n ∈ N.
It follows that for each n ∈ N, ν n ≤ µ n and ν n ≤ λ and, there is a subsequence ( N k j=0 |P j |) which converge almost everywhere to φ 0 and φ 0 is positive almost everywhere. Hence, by Cauchy criterium, for almost all z ∈ T,

+∞ k=n |P k (z)| ----→ n→∞ 1, that is, for almost all z ∈ T, φ n (z) ----→ n→∞ 1, (25)
and since (φ n (z)) n≥0 is bounded in L 2 (dz), we have that (φ n (z)) n≥0 is uniformly integrable in L 1 (dz), which implies, by Vitali Convergence Theorem (Theorem 4.5), that (φ n (z)) n≥0 converge in L 1 (dz) to 1. Obviously, from [START_REF] Flajolet | Analytic Combinatorics[END_REF], for almost all z ∈ T,

1 φ n (z) ----→ n→∞ 1, ( 26
) again we have that 1 φ n (z) n≥0 is uniformly integrable in L 1 (φ 2 0 dz) = L 1 (dµ). Indeed, ˆ1 φ 2 n (z) φ 2 0 dz = ˆ1 φ 2 n (z) dµ = ˆ1 φ 2 n (z) Q 2 n dµ n = ˆQ2 n dz = 1. and this gives that 1 φ n (z) n≥0 converge to 1 in L 1 (φ 2 0 dz), hence 1 φ n (z) n≥0 converge to 1 in L 1 (ν 0 ), since ˆ 1 φ n (z) -1 dν 0 ≤ ˆ 1 φ n (z) -1 dµ. ( 27
)
We thus get that the sequences (φ n (z)) n≥0 and 1 φ n (z) n≥0 converge to 1 in L 1 (ν 0 ). Whence, by virtue of Lemma 5.6, the Mahler measure of a subsequence (φ ni (z)) i≥0 converge to 1, and by Theorem 3.6, we have

+∞ j=ni M (P j ) ----→ i→+∞ 1.
We thus get that e is positive, since for any positive integer n, we have

M dµ dz = n-1 0 M (P j ).M dµ n dz .
It is remains to prove the second part of the theorem. Suppose µ and λ are mutually singular. Then, by Lemma 4. We shall apply the same reasoning as before. Since Q n converge to φ 0 in L 1 (dz), it follows easily that Q n converge to φ 0 in L 1 (τ ) and there is a subsequence Q ni such that, for almost all z ∈ T (with respect to τ ), by LlogL Zygmund theorem (Theorem 2.5) . This forces dµ dz to be a non zero constant function a.e., and yields that +∞ j=0 exp ˆlog P j (z) dz = e > 0, which is impossible in view of (⋆) combined with Theorem 3.6. Thus τ ≡ 0, that is, µ ⊥ λ, and this finishes the proof of the theorem.

Remark 5.7. By a standard argument from spectral analysis [52, pp. 17 2) Notice that it is easy to see that for any χ ∈ L 2 (X),

ˆχξ n dP -----→ n→+∞ 0.

Indeed, assume that χ is the indicator function of some set, then We conclude by the density of the simple functions in L 2 (X). An alternative proof can be obtained with the help of Lebesgue density theorem.

From Theorem 3.2 the proof of Theorem 3.1 is straightforward. Indeed, Proof of Theorem 3.1. By Theorem 3.2, the spectral type of any rank one is either singular or its absolute continuous part is equivalent to the Lebesgue measure according as the product of the Mahler measure of P j converge or diverge. Precisely, it is singular if and only if the product diverge otherwise the absolute continuous part is equivalent to the Lebesgue measure. The proof of the theorem is complete.

Let us prove Corollaries 3.5 and 3.7.

Proof of Corollaries 3.5 and 3.7. Straightforward, by Theorem 3.2 combined with Theorem 3.4.

Theorem 2 . 2 (

 22 Szegö, Kolmogorov-Krein [34, p.49], [29, p.136].

Theorem 2 . 3 (

 23 Kolmogorov [31, p.49].

Proposition 2 . 8 .

 28 The sequence n k=0 |P k (z)| converge weakly in L 2 (dz) to dµ dz . Let us point out that the rank one maps arising from the generalized Riesz product µ = k≥0 |P k | 2 and for any subsequence N ⊂ N , let us denote by ν the generalized Riesz product over the subsequence N , that is, the measure obtain as weak limit of the sequence of measures k∈N k≤n |P k (z)| 2 dz.

Theorem 3 . 3 (

 33 Third main result). Let ν = +∞ j=0 P nj 2 be a generalized Riesz product. Then ˆdν dz dz < 1.

Theorem 3 . 6 (

 36 [START_REF] El Abdalaoui | Calculus of generalized Riesz products[END_REF]). If each P k has less than c.h k-1 zeros bigger than 1 in absolute value where c is a positive constant less than one, then M dµ dz = 0.

Proof of Proposition 5 . 1 .P

 51 Let N be a positive integer and h be a continuous positive function. Then, By Cauchy-Schwarz inequality, we haveˆ k∈N , k≤N h P k (z) dz 2 ≤ ˆh P k0 (z) dz ˆh P k0 (z) k∈N \{k 0 }, k≤N P k (z) 2 dz .By letting N goes to infinity, from Proposition 2.8, we deduce that ˆh k0 (z) dz, and this proved the proposition.

3 .

 3 Proof of Theorem 3.3. By Proposition 5.1 we have

i≥0 1 f ni 1 <

 1 K ′ . Thus, by Vitali Convergence Theorem, it follows that log + f ni and log -f ni converge to 0 in L 1 (τ ), and this givesˆ log f ni dτ ---→ i→∞ 0,which yields that the Mahler measure of the sequence (f ni ) converge to 1, that is,

M

  (P j ) = 0 = e.For the converse, suppose that e = 0 and µ ⊥ λ. Then, there exists a positive measure τ such that τ ≤ µ and τ ≤ λ. Notice that we further haveMdµ n dz = 0, for all n ∈ N,

ni- 1 j=0P 1 . 1 .

 111 j (z) ----→ i→+∞ φ 0 (z). Hence, for almost all z ∈ T (with respect to τ ) Moreover the sequences φ ni and 1 φ ni are uniformly integrable in L 1 (τ ), since τ ≤ λ and τ ≤ µ. Therefore, by Vitali Convergence Theorem, φ ni and 1 φ ni converge in L 1 (τ ) to 1. Hence, by Lemma 5.6, we have exp ˆlog +∞ j=ni P j (z) dτ ----→ i→+∞ Notice that the careful application of Lemma 5.6 yields that ˆlog + +∞ j=ni P j (z) dτ, and this gives that ˆlog + +∞ j=0 P j (z) dτ < +∞.We may also take τ = +∞ j=0 P j (z)|dz and repeated the same reasoning to get dt ∈ H 1 ,

1 ξ 0 -χ 2 , 1

 121 where H -1 is the closed subspace generated by T k ξ 0 , k ≤ -1 . Thus ,by Fenchel-Rockafellar duality theorem[16, pp. 15-17], we have is the orthogonal complement of H -1 . Assume that we have M dµ dz = 0. Then, Our proof yields that the spectral type of the associated rank one is singular, we thus have H -1 = L 2 (X), where H -1 is the closed subspace generated by T k ξ 0 , k ≤ -1 and sup χ∈H -1 ⊥ ||χ||2≤1 ˆχξ 0 dP = 0.

  ˆχξ n dP ≤ |B n | -----→

Acknowledgment 1. The author wishes to express his heartfelt thanks to Mahendra Nadkarni who inspired largely this work and for his encouragements and many stimulating conversations. He would like also to express his thanks to François Parreau who introduced him to the Riesz products business and to Jean-Paul Thouvenot who introduced him to the wild world of rank one maps.


Jorge Luiz

Brito Gomes

Denise Maria Martins Vancea

Rodrigo Cappato De Araújo

Pooya Soltani

Fernando José De Sá

Pereira Guimarães

Manoel Da Cunha Costa

Rodrigo Cappato

Jorge Luiz De Brito Gomes

Fernando José

Sá Pereira Guimarães

Cardiovascular and enjoyment comparisons after active videogame and running in type-1 diabetics: A randomized

Keywords: videogames, diabetes mellitus, exercise, running, cardiovascular system, motivation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

de Brito Gomes, J. L., Martins Vancea, D. M., Cappato de Araújo, R., Soltani, P., de Sá Pereira Guimarães, F. J., da Cunha Costa, M. (Ahead of Print). Cardiovascular and enjoyment comparisons after active videogame and running in type-1 diabetics: A randomized crossover trial. Games for Health Journal. doi:10.1089/g4h.2020.0209

Introduction

Sedentary screen times, such as watching TV, playing video games, and using computers, as well as poor eating habits, may aggravate obesity, hypertension, and diabetes mellitus (DM) [START_REF] Brito-Gomes | Improving aerobic capacity through active videogames: A randomized controlled trial[END_REF][START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF] . Although type-1 DM (T1DM) affects approximately five percent of diagnosed diabetics, it should still be considered as an alarming disease because it could increase the risk of developing cardiovascular complications and mortality [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF]Standards of Medical Care in Diabetes-2011[END_REF][START_REF] Young | Good Cop, Bad Cop: Quality of Parental Involvement in Type 1 Diabetes Management in Youth[END_REF] . These complications include an increased heart rate (HR), systolic and diastolic blood pressure (SBP and DBP), and double product (DP), as well as changes in endothelial function (EF) [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF] Young | Good Cop, Bad Cop: Quality of Parental Involvement in Type 1 Diabetes Management in Youth[END_REF][START_REF] Hadi | Endothelial dysfunction: Cardiovascular risk factors, therapy, and outcome[END_REF] . Thus, the management of T1DM with diet, insulin, and exercise is important to reduce future cardiovascular risks [START_REF] Young | Good Cop, Bad Cop: Quality of Parental Involvement in Type 1 Diabetes Management in Youth[END_REF][START_REF] Hadi | Endothelial dysfunction: Cardiovascular risk factors, therapy, and outcome[END_REF] .

Although aerobic, strength, and combined exercises have been recommended as nonmedication treatments [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF]Standards of Medical Care in Diabetes-2011[END_REF][START_REF] Young | Good Cop, Bad Cop: Quality of Parental Involvement in Type 1 Diabetes Management in Youth[END_REF][START_REF] Faulkner | Personalized exercise for adolescents with diabetes or obesity[END_REF] , they may not be very enjoyable to practice and patients often lack adherence and motivation [START_REF] Medina | The Diabetic Adolescent and Sports Activity. International symposium on diabetes, education in health and oriented physical activities[END_REF] . In the recent years, active video games (AVG) have been used to reduce sedentary behaviors [START_REF] Brito-Gomes | Improving aerobic capacity through active videogames: A randomized controlled trial[END_REF][START_REF] Soltani | Sport exergames for physical education[END_REF][START_REF] Da | Do active and non-active videogames promote cardiovascular benefits during and after practice?[END_REF] . Being played primarily by healthy younger people [START_REF] Brito-Gomes | Improving aerobic capacity through active videogames: A randomized controlled trial[END_REF][START_REF] Soltani | Sport exergames for physical education[END_REF][START_REF] Da | Do active and non-active videogames promote cardiovascular benefits during and after practice?[END_REF][START_REF] Hart | Effects of video games on adolescents and adults[END_REF][START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF] , AVGs can acutely increase HR, BP, and DP, and reach the metabolic equivalents (METs) intensities recommended by the exercise guidelines [START_REF] Brito-Gomes | Improving aerobic capacity through active videogames: A randomized controlled trial[END_REF][START_REF] Da | Do active and non-active videogames promote cardiovascular benefits during and after practice?[END_REF][START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] Hurkmans | Energy expenditure in chronic stroke patients playing Wii Sports: a pilot study[END_REF][START_REF] Smallwood | Physiologic responses and energy expenditure of kinect active video game play in schoolchildren[END_REF] . Compared to traditional exercises, such as treadmill running, AVGs seem to be more enjoyable to practice [START_REF] Soltani | Effects of exergame and music on acute exercise responses to graded treadmill running[END_REF] .

Exercising with AVGs have also shown clinical benefits for DM patients, including reductions in BMI, improvements in glycated hemoglobin, muscle strength, walking ability, and metabolic control [START_REF] Faulkner | Personalized exercise for adolescents with diabetes or obesity[END_REF][START_REF] De Brito-Gomes | Exercises in physically active screen time: can active videogames be one tool in health control of type 1 and 2 diabetes?[END_REF] . However, it is still not clear which exercises (traditional or non-traditional) handle improvements in T1DM. It is also important to explore whether T1DM could have cardiovascular risks in the future if not handled correctly. Finally, although one single exercise session is important, other practices, including AVG, may also be enjoyable and present cardiovascular benefits in T1DM. Therefore, the aim of current study was to compare the effects of an AVG and traditional exercise session (treadmill running) on cardiovascular and enjoyment responses of T1DM patients. Thinking about the new concepts of the literature about comparisons of non-inferiority clinical studies of one intervention with another [START_REF] Hahn | Understanding noninferiority trials[END_REF][START_REF] Head | Non-inferiority study design: lessons to be learned from cardiovascular trials[END_REF] , we hypothesize that both sessions will present similar cardiovascular responses, and that patients in AVG session will show higher enjoyment.

Material and methods

Trial design

This is a prospectively registered (UTN: U1111-1194-370), randomized clinical trial with cross-over design. The study was approved by the local research ethics committee (Number: 1.560.045), and all participants gave their written informed consent prior to the assessments.

Participants and eligibility criteria

We recruited T1DM patients by social media and through the health campus of Pernambuco University. Participants were included if they met the following criteria: 1) female and male T1DM patients; 2) regularly using insulin; 3) do not use anxiolytics, anti-inflammatories, antibiotics, or beta-blockers; 4) do not have bone-myoarticular restriction (e.g., amputations, unhealed wounds, and adhesive capsulitis), peripheral neuropathy or any impairment and complication caused by diabetes that prevents them from participation in the sessions. Participants were excluded if they: 1) performed other physical exercises 24 hours before or after the sessions; 2) encounter any type of complication outside the research that was aggravated by the practice of the sessions; and 3) by medical report, requesting the absence or exit from the study. The following recommendations were made selecting the participants and before their evaluation: 1) to wear minimal clothing specific to exercise; 2) to avoid moderate or vigorous exercise for 24 hours prior to testing; and 3) to refrain from smoking, alcohol, coffee, or any other stimulants that could interfere with cardiovascular responses.

Sample size

A posteriori sample size calculation was estimated by G*Power 3.1.9 software [START_REF] Faul | Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses[END_REF] , given α= 0.05, power (1-β) = 0.8, and using effect size (ES) for HR, SBP, DBP, DP, VD, and %EF. Considering 3 groups and 4 repeated measures (pre, immediately after, 30 minutes after, and 24 hours after), a minimum of 9 participants per group was required to conduct this study.

Randomization

The randomization method was performed by allocation using a website randomizer to propose the participant's order with 50% of the volunteers starting with the active AVG session and 50% starting with running for each session (ratio 1:1). Thus, after the control session, each pair of participants were allocated from the order in which the procedures began, one would start with either AVG or treadmill running. Two-by-two volunteer's allocation were performed to better control the measurement.

Study design

The present study consisted of twice a week visits of the Human Performance Assessment Laboratory (temperature 22-26 ºC; ambient relative humidity 40%-60%; atmospheric pressure ~ 101.3 Pa; Figure 1).

Week 1 (days 1 and 2)

On day 1, the participants underwent anthropometric and body composition assessments. After that, they remained at rest for cardiovascular measurements, including HR, SBP, DBP, DP, EF, and blood glucose (G). Next, they performed the Control session in the sitting position with direct measurement of gases during the 30 minutes. All the measurements, except gas analysis, were reevaluated immediately after, 30 minutes after, and 24 hours after the session in the same sitting position. On the second day, blood sample, HR, SBP, DBP, DP, and EF measurements were retaken. After, the maximum oxygen consumption test (VO2max test) was condicted. Then, the randomization of sessions was performed using allocation [START_REF] Ferreira | Randomization: More than the tossing a coin[END_REF] .

Week 2 (day 3 and 4)

On the second week, to avoid residual fatigue, and after an interval of 72-196 hours, the first active session was realized. On the third day, the volunteers remained at rest for 10 minutes and after that, HR, SBP, DBP, DP, EF and G measurements were taken in seated position. Depending on the allocation, the trial consisted of 30 minutes of playing with Microsoft Xbox 360 Kinect Adventures AVG or running on the treadmill. During each session, cardiovascular (HR) and metabolic (MET) variables were recorded. Immediately after the session, 30 minutes and 24 hours (fourth day), the primary outcomes (cardiovascular and enjoyment) were reevaluated and secondary outcomes (MET and G) were evaluated under the same conditions (Figure 2).

Week 3 (Day 5 and 6)

Depending on the activity performed in the second week, all procedures were methodologically repeated in the third week for the the other active session (cross-over active sessions).

Measures

Primary outcomes (cardiovascular and enjoyment)

Cardiovascular: After 5 minutes of rest, HR (bmp) was measured for 10 minutes using an HR monitor (RS800 CX; Polar Electro Oy, Finland). SBP (mm Hg) and DBP (mm Hg) were measured on the left arm using a blood pressure monitor (HEM 7113; Omron Healthcare, Japan). DP was calculated as SBP × HR (mm Hg × bpm) for the same period, providing valuable information on the cardiac workload and myocardial oxygen demand. All cardiovascular measures were taken in the sitting position, and the participants were asked to remain silent.

For EF (vessel diameter-VD and %EF), images of the brachial artery were captured using high-resolution ultrasound. This allowed analyzing blood flow-mediated dilation after 4.5 minute occlusion of the brachial artery with an inflated aneroid cuff (1.5 × SBP, measured after the 10minute rest of cardiovascular measures) [START_REF] Celermajer | Non-invasive detection of endothelial dysfunction in children and adults at risk of atherosclerosis[END_REF][START_REF] Steyers | Endothelial dysfunction in chronic inflammatory diseases[END_REF] . If there is less than 10% dilation at baseline, it is considered as endothelial dysfunction. To avoid any types of bias in the analysis of EF, participants' right arms were standardized, and baseline blood pressure measurements and laboratory blood measurements were performed on the left arm.

Knowing the possible measurement error in the analysis, initially a study of reproducibility of the measurement of EF by ultrasound was performed with non-diabetic people, presenting an intraclass correlation coefficient with values considered excellent (> 0.90) [START_REF] Koo | A guideline of selecting and reporting intraclass correlation coefficients for reliability research[END_REF] . We observed values of 0.902 (95% CI; 0.752 -0.961) for vessel dilation at the pre-resting moment (day 1 vs. day 2), and 0.967 (95% CI; 0.915 -0.987) for vessel dilation at the time after 60 seconds. At rest, we observed values of 0.922 (95% CI; 0.802 -0.969) for vessel dilation before physical exercise (day 1 vs. day 2), and 0.940 (95% CI; 0.846 -0.976) for dilation of the vessel at the moment after 60 seconds of physical exercise (day 1 vs. day 2). Technical measurement errors were also low (< 5%) [START_REF] Perini | Calculation of technical error of measurement in anthropometry[END_REF][START_REF] Diniz | Evaluation of ultrasound reproducibility as a method of measurement of abdominal and visceral fat[END_REF] , with values of 3.56%, 2.97%, 2.66%, and 2.73%, respectively to those mentioned above.

Enjoyment level: A 100-mm visual analog scale, with reproducibility of 0.89, in the paper-and-pencil form, ranging from 0 (very boring) to 10 (very enjoyable) centimeters, was used to measure enjoyment immediately after the sessions [START_REF] Ahearn | The use of visual analog scales in mood disorders: A critical review[END_REF][START_REF] De Brito-Gomes | Physical Effort, Energy Expenditure, and Motivation in Structured and Unstructured Active Video Games: A Randomized Controlled Trial[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] .

Secondary outcomes (physical effortcardiovascular and metabolic)

Cardiovascular: Average and maximal HR was verified during the session. Metabolic: Maximum oxygen consumption (VO2max) was measured during the initial assessment using a computerized metabolic analyzer (Quark Cosmed CPET, Minnesota, Germany), and based on a treadmill test. The protocol consisted of a two-minute warm-up at 5 km/h, after which the intensity was continuously increased by 1 km/h until the maximum voluntary fatigue. The highest VO2 value found before the end of the test was noted as participant's oxygen consumption (VO2peak).

Blood glucose: Cappilary blood glucose (mg•dL -1 ) was collected using a clinical glucometer (Accu-Check Active, Roche, Brazil), that assessed glycemic levels in approximately five seconds, and a lancet device, disposable test strips, and disposable lancets (Accu-Check Softclix, Roche, Brazil). Blood glucose was assessed before, immediately after, and 30 minutes after each session to verify the session's glycemic safety. Recommended blood glucose level (higher than 100 and lower than 300 mg•dL -1 ) was used to start the sessions and was suspended with symptoms of hyperglycemia (> 250 -300 mg•dL -1 ) or hypoglycemia (< 100 mg•dL -1 ). In cases of exercise suspension, we would release T1DM only after their safe glycemic correction (eg. carbohydrate supplementation).

Interventions

Microsoft Xbox 360 Kinect game console (Microsoft, USA) and Kinect Adventures! game (Microsoft Game Studios, USA) were used [START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] . Unpublished data from our lab verified the intensity of all mini-games of Kinect Adventures! Previous research has also verified the moderate intensity of River Rush, Rally Ball, and Reflex Ridge mini-games in 14 non-diabetics adult volunteers when played in all-out performance [START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] . T1DM patients played three most intense games of Kinect Adventures!, including River Rush, Rally Ball, and Reflex Ridge, for 10 minutes each (30 minutes total) and with all-out intensity. The goal of River Rush game was to gather as many pins as possible in the water and air. In Rally Ball game, participants had to deflect balls and hit them down the court, and at the same time collect the balls and prevent them from passing. Reflex Ridge game asked players to race on a platform, jump over hurdles, lean away from obstacles to avoid hitting their heads on low beams. In all mini-games, participants performed jumps, squats, and lateral shifts with vertical and horizontal shoulder extension for collecting all possible pins.

The running session consisted of running on the treadmill with moderate intensity (45-59% VO2max) according to recommendations [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF]Standards of Medical Care in Diabetes-2011[END_REF][START_REF] Young | Good Cop, Bad Cop: Quality of Parental Involvement in Type 1 Diabetes Management in Youth[END_REF][START_REF] Garber | American College of Sports Medicine position stand. Quantity and quality of exercise for developing and maintaining cardiorespiratory, musculoskeletal, and neuromotor fitness in apparently healthy adults: guidance for prescribing exercise[END_REF] , with one minute in each intensity and a total duration of 30 minutes. The maximum oxygen consumption achieved during the maximum test, was converted to METs and was used to determine the speeds of 45 and 59% VO2 [START_REF] Garber | American College of Sports Medicine position stand. Quantity and quality of exercise for developing and maintaining cardiorespiratory, musculoskeletal, and neuromotor fitness in apparently healthy adults: guidance for prescribing exercise[END_REF][START_REF]ACSM Guidelines for Exercise Testing and Prescribing[END_REF] . A three-minute warm-up was also performed at 4-5 km/h to start the activity.

Statistical analysis

The data were analyzed using the Prisma program (GraphPad version 3.0, San Diego, USA). A priori, an exploratory analysis of normality (Shapiro-Wilk test), and descriptive measures (mean and standard deviation) were performed. For intragroup and intergroup analysis, two-way analysis of variance with repeated measures and with post hoc Tuckey was performed to check the cardiovascular and blood glucose changes [Group vs. Time] during pre-, during, and after sessions (30 minutes and 24 hours). One-way analysis of variance was performed to compare the metabolic equivalents, maximal, and average heart rate and enjoyment levels during the sessions. The level of significance was set to p < 0.05. The effect size (ES) was also calculated using the following formula for paired samples, repeated measures, and pre-and post-intervention [START_REF] Field | Discovering statistics using SPSS[END_REF][START_REF] Espirito-Santo | Calculating and presenting effect sizes in scientific works (1): The limitations of p <0.05 in the analysis of mean differences between two groups[END_REF] :

√ 𝐹 (𝑑𝑓) ____ 𝐹(𝑑𝑓) + 𝑑𝑓
where "F" is the Anova value and "df" is the degrees of freedom (df = n-1). ES scores of 0.01-0.20 was considered small, 0.21-0.50 moderate, 0.51-0.80 large, and ≥0.80 was considered as very large. The minimum detectable difference (MDD) was also calculated to verify the clinical benefits of the variables using the following formula [START_REF] Alves | Tai-chi-chuan and yoga on post exercise hypotension: comparison to aerobic and resistance exercise[END_REF] :

MDD = (SD × √1 -ICC) × 1.96 × √2
where SD is standard deviation and ICC is intraclass correlation coefficient.

Results

Participants flow diagram

There was no dropout according to the pre-established exclusion criteria throughout the randomized clinical trial. The eligibility setting is shown in figure 3. The descriptive information of participants is presented in table 1.

Table 1. Baseline descriptive information (n=10).

Mean (standard deviation) Age (years) 24.9 (7. Primary and secondary outcomes are presented in table 2. Statistically significant cardiovascular changes were found among the groups. HR, SBP, and DP statistically changed immediately after and 30 minutes after active sessions (p < 0.001) compared to the control session. DBP statistically changed only 30 minutes after the AVG session (p = 0.016) compared to the control session. AVG also showed statistical changes compared to the running session in HR, SBP, DBP, and DP, immediately after and 30 minutes after the activity. VD and %EF were statistically changed 30 minutes and 24 hours after active sessions (p < 0.001) compared to the control session. Statistically significant enjoyment levels were found among the groups and AVG session showed higher values compared to the running and control sessions (p < 0.001). Statistically significant changes were also observed in blood glucose, HR and, METs compared to a control session with no statistical differences between active sessions. 9) 83 ( 9) 82 ( 9) 80 ( 8) 82 ( 9) 129* (13) 94* ( 8) 78 ( 10) 82 ( 9) 142* # (13) 98* ( 12) 75 ( 8) SBP (mmHg) 118 ( 8) 120 ( 7) 118 ( 11) 118 ( 7) 118 ( 6) 129* ( 13) 119 ( 10) 118 ( 7) 116 ( 8) 130* (15) 113 # ( 10) 115 ( 9) DBP (mmHg) 70 ( 5) 74 ( 5) 75 (10) 67 ( 6) 69 ( 7) 77 ( 6) 77 ( 5) 68 ( 6) 67 ( 5) 77 ( 5) 70* # ( 8) 66 ( 4)

DP

Discussion

The aim of this randomized clinical trail was to compare the effects of AVG and traditional exercise sessions on cardiovascular and enjoyment responses of T1DM patients.

Higher responses compared to control and rest

Our sessions resulted in changes in cardiovascular variables immediately after the exercise compared to the rest or control sessions. HR and SBP had statistically significant increase immediately after active sessions. Similar responses were observed in the DP but without exceeding the maximum allowed values, maintaining the hemodynamic safety of the participants (< 30 mmgH.bpm) [START_REF] Alves | Tai-chi-chuan and yoga on post exercise hypotension: comparison to aerobic and resistance exercise[END_REF][START_REF] Fornitano | Double high product as a predictor of the absence of graded obstructive coronary artery disease important in patients with a positive exercise test[END_REF] . Additionally, our sessions showed similar changes without statistical significance for HR, BP, and DP variables after 30 minutes and 24 hours after exercise compared to rest or control sessions. It is possible that volunteers' normotensive baseline values have contributed to a non-significant postexercise hypotension (HPE) after a single session [START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] Meneses-Junior | Hemodynamic responses during and after a Pilates session compared to aerobic and resistance exercise[END_REF][START_REF] Brito-Gomes | Active videogames promotes cardiovascular benefits in young adults? Randomized controlled trial[END_REF] . We also observed that active sessions promoted similar increase in VD and %EF values after 30 minutes. These values remained high after 24 hours compared to rest and control sessions. This could have happened because of the reduced myocardial effort, that resulted in a reduction of peripheral vascular resistance after AVG and running sessions compared to the rest or the control group [START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] .

Regarding blood glucose, AVG and running sessions promoted reductions immediately after and 30 minutes after the sessions compared to the rest or control group. However, the enjoyment levels were significantly higher only in AVG session compared to the control group. It has been shown that AVG audiovisual feedback could increase enjoyment levels [START_REF] Soltani | Effects of exergame and music on acute exercise responses to graded treadmill running[END_REF] , and our data supported these findings in T1DM patients after an AVG session. It should be noted that no statistical differences were identified between the running and control sessions. This shows how discouraging it might be to practice traditional running exercises for T1DM patients. Some volunteers reported they practiced traditional exercises because they knew it was necessary, but they did not like it as much as the AVG session.

Comparison between AVG and running

We found similarities between the immediate and late acute cardiovascular responses between the AVG and treadmill running sessions. Regarding HR, BP, and DP cardiovascular outcomes, T1DM patients had similar increase but without statistically significant HPE after 30 minutes and 24 hours after the session. Similar behavior was observed in other types of light to moderate intensity activities [START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF][START_REF] Brito-Gomes | Active videogames promotes cardiovascular benefits in young adults? Randomized controlled trial[END_REF] . For example, HR values returned to baseline levels after 15 minutes, without promoting hypotension over the time [START_REF] Brenton | Adding heart rate signal to a control-to-range artificial pancreas system improves the protection against hypoglycemia during exercise in type 1 diabetes[END_REF] . It seems that the intensity of the exercise plays a role in increasing HPE in T1DM patients compared to healthy adults [START_REF] Soltani | Effects of exergame and music on acute exercise responses to graded treadmill running[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] . We also found similar increase in VD and %EF, 30 minutes and 24 hours after the sessions. Our results corroborate the findings of Hall et al. [START_REF] Hall | Effects of hypoxia during continuous and intermittent exercise on glycaemic control and selected markers of vascular function in type 1 diabetes[END_REF] , in which an intermittent aerobic session can cause a vessel size relaxation over 100%. Physiologically, intermittent periods in a session with higher-intensity bouts could lead to an adrenalin release and improvements in vascular system reactivity in TD1M after exercise because of the vascular endothelial growth factor enhancement [START_REF] Hall | Effects of hypoxia during continuous and intermittent exercise on glycaemic control and selected markers of vascular function in type 1 diabetes[END_REF] .

On the other hand, we observed that SBP was reduced significantly 30 minutes after the AVG session compared to the running session of the same duration. One possible reason is that the participants have shown higher values of maximal HR during the AVG session, which could have led to increased VD and % EF values. Physiologically, higher reductions of peripheral vascular resistance were observed in AVG session, which caused reductions in the systolic cardiac ejection strength [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF] Celermajer | Non-invasive detection of endothelial dysfunction in children and adults at risk of atherosclerosis[END_REF][START_REF] Steyers | Endothelial dysfunction in chronic inflammatory diseases[END_REF] . Similarly, significant differences of DBP were found between sessions, with the AVG session showing reduced values in blood return strength after 30 minutes, as well as further reductions after 24 hours, compared to the control session.

It is important to note that the primary outcomes had very large ES values in HR, SBP, DP, VD, and %EF (ES > 0.8), and a large value in DBP (ES = 0.606). Our results were also higher than the ES values of T1DM studies with traditional aerobic and resistance exercises identified in a metaanalysis (ES = 0.12, when compared to the control group, and 0.51 < ES < 0.65, when compared to independent or pre-test groups) [START_REF] Nielsen | Meta-analysis of the effect of exercise interventions on fitness outcomes among adults with type 1 and type 2 diabetes[END_REF] . The larger ES could have happened because of the exercise prescription-evaluation and group sessions [START_REF] Nielsen | Meta-analysis of the effect of exercise interventions on fitness outcomes among adults with type 1 and type 2 diabetes[END_REF] . The present study was performed with a crossover randomized design with three sessions (two active sessions and one control group comparison) in repetitive measures over time which had improved the strength of the results.

Our findings on the capillary blood glucose presented similar reductions between the sessions without statistical differences immediately after-and 30 minutes after the exercises. Previous studies evaluated the effect of AVG in non-diabetics and found different responses, possibly due to different exercise intensities [START_REF] Soltani | Effects of exergame and music on acute exercise responses to graded treadmill running[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] . However, these reductions are important metabolic responses to control the illness under real-life conditions. These values can also occur after exercise because of the glucose consumption, exercise sessions over than 60 minutes, or even exercise intensity [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF]Standards of Medical Care in Diabetes-2011[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] . Our study supports safe reductions for T1DM patients after 30 minutes moderate-intensity exercises without statistically significant hypoglycemia (< 70 mg. dL -1 ). These values are important for maintaining good disease management and to reduce future cardiovascular and metabolic risks [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF]Standards of Medical Care in Diabetes-2011[END_REF] .

We also observed very large ES values for the enjoyment levels with statistical differences between active sessions. Enjoyment levels were also higher during the AVG session compared to the running session, suggesting that AVG is more motivating for T1DM patients. Previous study evaluated the effect of the same AVG (Kinect Adventures) in non-diabetics, and found that this game could present greater enjoyment levels with human-computer-exercise interaction [START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] . Our data supported these findings in T1DM patients, while comparing it with a traditional exercise session.

Clinical relevance, the novelty, and study limitations

Besides the statistical analysis, the cardiovascular clinical data from MDD was verified. HR, BP, DP, and EF were clinically relevant in AVG and running sessions immediately after, as well as 30 minutes and 24 hours after the sessions. These results could prevent atherosclerotic denegation, inflammatory responses, and future coronary risks [START_REF] Society | Brazilian Diabetes Society Guidelines[END_REF][START_REF]Standards of Medical Care in Diabetes-2011[END_REF] . Previous studies have also verified similar MDD results after AVG and traditional exercise sessions, and have shown that minimum MDD changes could promote cardiovascular and metabolic health [START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF][START_REF] Alves | Tai-chi-chuan and yoga on post exercise hypotension: comparison to aerobic and resistance exercise[END_REF] . Therefore, health professionals who work with T1DM patients can use our results for including AVGs as complementary management tools that provide cardiovascular benefits with great enjoyment levels. The novelty of this study was that it verfied the acute and late cardiovascular responses and the enjoyment levels of AVG and traditional exercise sessions in T1DM patients. Our data differed from a) previous studies performing AVG and running in non-diabetics, which only considered the immediate acute physiological changes [START_REF] Hart | Effects of video games on adolescents and adults[END_REF][START_REF] Hurkmans | Energy expenditure in chronic stroke patients playing Wii Sports: a pilot study[END_REF][START_REF] Smallwood | Physiologic responses and energy expenditure of kinect active video game play in schoolchildren[END_REF][START_REF] Soltani | Effects of exergame and music on acute exercise responses to graded treadmill running[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF] ; b) studies performing only with AVG without traditional exercise comparison in non-diabetics [START_REF] Hart | Effects of video games on adolescents and adults[END_REF][START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] Hurkmans | Energy expenditure in chronic stroke patients playing Wii Sports: a pilot study[END_REF][START_REF] Soltani | Effects of exergame and music on acute exercise responses to graded treadmill running[END_REF][START_REF] De Brito-Gomes | Physical Effort, Energy Expenditure, and Motivation in Structured and Unstructured Active Video Games: A Randomized Controlled Trial[END_REF][START_REF] De Brito-Gomes | Do 30 minutes of active video games at a moderate-intensity promote glycemic and cardiovascular changes?[END_REF][START_REF] Brito-Gomes | Active videogames promotes cardiovascular benefits in young adults? Randomized controlled trial[END_REF] ; c) the study with AVG and traditional exercise modalities in the same training program which does not allow verifying what exercise type was handled metabolic improvements in T1DM 6 ; and d) studies verifying non-diabetics cardiovascular responses after AVG without considering enjoyment levels [START_REF] Hart | Effects of video games on adolescents and adults[END_REF][START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] Hurkmans | Energy expenditure in chronic stroke patients playing Wii Sports: a pilot study[END_REF][START_REF] Smallwood | Physiologic responses and energy expenditure of kinect active video game play in schoolchildren[END_REF][START_REF] Brito-Gomes | Active videogames promotes cardiovascular benefits in young adults? Randomized controlled trial[END_REF] .

There were some limitations to this study. First, we performed our study with a single AVG. Although the study showed late acute responses of Kinect Adventures AVG compared with traditional exercise, future studies should verify the benefits of different AVGs in T1DM compared to traditional exercises [START_REF] Brito-Gomes | Improving aerobic capacity through active videogames: A randomized controlled trial[END_REF][START_REF] De Brito-Gomes | Does a virtual functional training induce cardiovascular responses in normotensive adults after a single session and over weeks?[END_REF][START_REF] De Brito-Gomes | Exercises in physically active screen time: can active videogames be one tool in health control of type 1 and 2 diabetes?[END_REF][START_REF] De Brito-Gomes | Physical Effort, Energy Expenditure, and Motivation in Structured and Unstructured Active Video Games: A Randomized Controlled Trial[END_REF] . Second, a young adult cohort was used. Although there were no age-sex differences in our study and other similar studies with mixed male-female T1DM participants 6,35- 39 , future different age cohort should be studied. Sex-group comparisons with higher sample sizes should also be performed to verify possible age and sex differences of T1DM patients during AVG and traditional exercises.

Conclusion

The aim of this study was to compare the cardiovascular and enjoyment responses of AVG vs. traditional exercise in T1DM patients. Our results showed similar responses between AVG and the running session in majority of the cardiovascular responses. However, AVG showed an increase in EF with greater magnitudes values after exercise and compared to the running session. Finally, the enjoyment levels of T1DM patients were higher during AVG compared to the running session.

Figure 1 .

 1 Figure 1. Study design.

Figure 2 .

 2 Figure 2. Experimental session design. DBP: Diastolic blood pressure; EF: Endothelial function; G: Blood glucose; HbA1c: Glycated hemoglobin; HR: Heart rate; EL: Enjoyment level; METs: Metabolic equivalents; SBP: Systolic blood pressure.

Figure 3 .

 3 Figure 3. The flow diagram of study participants.

  VO2 Peak (ml/kg/min) 37.4 (6.6) NPH: Neutral Protamine Hagedorn; HbA1C: Glycated hemoglobin type A1C.

Note 1 :

 1 The following effect size (ES) are due to the interaction (group × time): HR ES = 2.71, p < 0.001; SBP ES = 0.879, p < 0.001; DBP ES = 0.606, p = 0.016; DP ES = 2.282, p < 0.001. VD ES = 1.871, p < 0.001. EF ES = 1.165, p < 0.001, Enjoyment ES = 1.74, p < 0.001. Note 2: MDD HR = 4 bpm, MDD SBP = 5 mm.Hg, MDD DBP = 6 mm.Hg, MDD DP = 1310 bpm × mmHg, MDD VD = 4.4 mm, and MDD %EF = 8.2%.

Table 2 .

 2 Cardiovascular and enjoyment responses presented as mean (standard deviation).

			Control session			Running session			AVG session
		B	IA	30 min	24h	B	IA	30min	24h	B	IA	30 min	24h
	HR (bpm)	79 (

  Immediately after exercise; 30 min: 30 minutes after exercise; 24h: 24 hours after exercise; HR: Heart rate a rest; SBP: Systolic blood pressure at rest; DBP: Diastolic blood pressure at rest; DP: Double product; VD: Vessel diameter; EF: % of endothelial function. *p < 0.05 in relation to the control session. #p < 0.05 in relation to the running session.

	(mmHg.bpm)	9145	9775	9478	9232	9791	17150*	11167*	9104	9373	18540*	10944*	8518
		(487)	(816)	(1161)	(708)	(1091)	(2861)	(1276)	(922)	(757)	(3027)	(1499)	(474)
	VD (mm)	39.8 (10.9)	-	40.5 (10,1)	38,7 (10.3)	41.5 (9.9)	-	47.4* (10.1)	46.4* (12.4)	39.6 (9.5)	-	48.8* (12.3)	56.6* # (13.9)
	EF (%)	7.9 (9.2)	-	3.7 (11.1)	7.4 (11.8)	7.3 (9.4)	-	14.8* (14.1)	26.8* (18.9)	9.6 (8.5)	-	29.6* # (17.1)	45.4* # (25.9)
	Enjoyment (units)	-	5.6 (2.3)	-	-	-	7.7 (1.6)	-	-	-	9.4 (0.7)* #	-	-
							Secondary outcomes
	Blood glucose	224 (47)	231 (40)	231 (66)	-	168 (30) 109* (40) 140* (54)	-	179 (60) 138* (69) 152* (67)	-
	Average HR (bpm)		83 (11)			133 (14)*			132 (12)*
	Maximal HR bpm)		104 (14)			154 (19)*			176 (21)*
	MET (units)		1.13 (0.2)			4.6 (1.1)*			4.0 (0.8)*
	B: Baseline; IA: