Printer Friendly

On ergodic measures with negative Lyapunov exponents.

1. Introduction. It is well known that every ergodic measure whose Lyapunov exponents are all negative of a [C.sup.1+[alpha]] diffeomorphism is supported on an attracting periodic orbit (cf. Corollary S.5.2 in [6]). This result was extended to [C.sup.1] diffeomorphisms by Araujo [1]. In the case of flows, Campanino [3] proved that every nonatomic ergodic measure whose Lyapunov exponents off the flow direction are all negative of a [C.sup.1+[alpha]] flow is supported on an attracting periodic orbit. The methods in [10] imply that this is also true in the [C.sup.1] class but for star flows, i.e., flows which cannot be [C.sup.1] approximated by ones with nonhyperbolic critical elements.

In this paper we will extend Campanino's result to general [C.sup.1] n-dimensional flows with n [greater than or equal to] 3. More precisely, we will prove for all such flows that every nonatomic ergodic measure whose Lyapunov exponents off the flow direction are all negative is supported on an attracting periodic orbit. Let us state our result in a precise way.

Hereafter the term n-dimensional flow will mean a [C.sup.1] vector field X defined on a compact connected boundaryless Riemannian manifold M of dimension n [member of] [N.sup.+]. The one-parameter group of diffeomorphisms generated by X will be denoted by [X.sub.t], t [member of] R. We say that x [member of] M is a periodic point of X if there is a minimal positive number ^(x) (called period) such that [X.sub.[pi](x)] (x) = x. Notice that if x is a periodic point, then 1 is an eigenvalue of the derivative [DX.sub.[pi](x)] (x) with eigenvector X(x). The remainders eigenvalues of [DX.sub.[pi](x)] (x) will be referred to as the eigenvalues of x. We say that a periodic orbit O(x) = {[X.sub.t](x) : t [member of] R} is attracting if every eigenvalue of x has a modulus less than 1.

Let [mu] be a Borel probability measure of M. We say that [mu] is nonatomic if it has no points with positive mass. We say that [mu] is supported on H [subset] M if supp([mu]) [subset] H, where supp([mu]) denotes the support of [mu]. We say that [mu] is invariant if [mu]([X.sub.t](A)) = [mu],(A) for every Borelian A and every t [member of] R. An invariant Borel probability measure is ergodic if every measurable invariant set has measure 0 or 1.

Oseledets's Theorem [11] ensures that every ergodic measure [mu] is equipped with a full measure set R, a positive integer k and real numbers [[chi].sub.1] < [[chi].sub.2] < ... < [[chi].sub.k] such that for every x [member of] R there is a measurable splitting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for every x [member of] R, [e.sup.i] [member of] [E.sup.i.sub.x]\{0}, 1 [less than or equal to] i [less than or equal to] k. Such numbers are so-called the Lyapunov exponents of [mu]. Similar definitions and results hold for [C.sup.1] diffeomorphisms.

With these definitions we can state our result.

Theorem 1. Let [mu] be a nonatomic ergodic measure of an n-dimensional flow with n [greater than or equal to] 3. If the Lyapunov exponents of [mu] off the flow direction are all negative, then [mu] is supported on an attracting periodic orbit.

2. Proof. We divide the proof of Theorem 1 into three parts according to the following subsections.

2.1. Linear Poincare flow. Given a flow X we denote by Sing(X) the set of singularities (i.e. zeroes) of X. Define [M.sup.*] = M \ Sing(X) as the set of regular (i.e. nonsingular) points of X. To any x [member of] [M.sup.*] we define [N.sub.x] [subset] [T.sub.x] M as the set of tangent vectors which are orthogonal to X(x). Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the vector bundle so induced and, correspondingly, denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the orthogonal projection. Define the linear Poincare flow as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will need the following lemmas about [P.sub.t]. It is here where the hypothesis n [greater than or equal to] 3 is used in the proof of Theorem 1.

Lemma 2. For every n-dimensional flow X with n [greater than or equal to] 3 and every T [member of] R there exists K > 0 such that [parallel][P.sub.T](x)[parallel] > K for every x [member of] [M.sup.*].

Proof. Otherwise, there exists a sequence [x.sub.k] [member of] [M.sup.*] such that [parallel][P.sub.T]([x.sub.k])[parallel] [right arrow] 0 as k [right arrow] [infinity]. By compactness we can assume [x.sub.k] [right arrow] [sigma] for some [sigma] [member of] [M.sup.*]. Since [parallel][P.sub.T]([x.sub.k])[parallel] [right arrow] 0, [sigma] is a singularity and so [X.sub.T]([x.sub.k]) [right arrow] [sigma] too. Again by compactness we can assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some one-dimensional subspaces L and L' of [T.sub.[sigma]]M, where [E.sup.X] is the one-dimensional subbundle of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] generated by X. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges to the orthogonal complement N of L in [T.sub.[sigma]]M. Similarly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges to the orthogonal complement N' of L' in [T.sub.[sigma]]M. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the orthogonal projection. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we conclude that [P.sub.T]([x.sub.k]) [right arrow] [[pi].sub.N'] [omicron] [DX.sub.T]([sigma])[|.sub.N] as k [right arrow] [infinity]. Since [parallel][P.sub.T]([x.sub.k])[parallel] [right arrow] 0 as k [right arrow] [infinity], we obtain [[pi].sub.N'] [omicron] [DX.sub.T]([sigma])[|.sub.N] = 0 which is equivalent to [DX.sub.T]([sigma])N [subset] L'. However, n [greater than or equal to] 3 and dim(L) = 1, so dim(N) [greater than or equal to] 2, thus dim([DX.sub.T]([sigma])N) [greater than or equal to] 2. Since dim(L') = 1 and [DX.sub.T]([sigma])N [subset] L', we obtain a contradiction. This ends the proof.

Lemma 3. Let [mu] be a nonatomic ergodic measure of a flow X. If the Lyapunov exponents of [mu] off the flow direction are all negative, then there is [T.sub.0] > 0 such that p is an ergodic measure of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. First we prove [mu](Sing(X)) = 0. Otherwise, [mu](Sing(X)) = 1 since [mu] is ergodic and Sing(X) is closed invariant. Since every Lyapunov exponent of [mu] off the flow direction is negative, every Lyapunov exponent of [mu] is therefore negative (and so different from zero). In such a case, the results in p. 632 of [4] imply that [mu] is supported on a singularity. However, this contradicts that [mu] is nonatomic. Hence, [mu](Sing(X)) = 0.

Let us continue with the proof. Since [mu] is ergodic and [mu](Sing(X)) = 0, Oseledets's Theorem for the linear Poincare flow (cf. Theorem 2.2 in [2]) implies that there exist a full measure set R [subset] [M.sup.*], a [P.sub.t]-invariant splitting [N.sub.R] = [N.sup.1] [direct sum] ... [direct sum] [N.sup.p] and real numbers [[bar.[chi]].sub.1] < ... < [[bar.[chi]].sub.p] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for every x [member of] R, [v.sup.i] [member of] [N.sup.i.sub.x], 1 [less than or equal to] i [less than or equal to] p. Again by Oseledets's which is now applied to the flow X, we also have an invariant measurable splitting [E.sup.1] [direct sum] ... [direct sum] [E.sup.k] over R with Lyapunov exponents [[chi].sub.1] < ... < [[chi].sub.k] of [mu] as an ergodic measure of X. Since every Lyapunov exponent of [mu] off the flow direction is negative, [[chi].sub.k] = 0 and so [[chi].sub.k-1] < 0.

Take v [member of] [N.sup.i.sub.x] for some x [member of] R and 1 [less than or equal to] i [less than or equal to] p. Write v = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some J [subset] {1, ..., k - 1} and [v.sub.j] [member of] [E.sup.j.sub.x] for all j [member of] J. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Letting t [right arrow] [infinity] we get [[bar.[chi]].sub.i] [less than or equal to] max{[[chi].sub.1], ..., [[chi].sub.k-1]} = [[chi].sub.k-1] < 0. Hence the numbers {[[bar.[chi]].sub.1], ..., [[bar.[chi]].sub.p]} are all negative too.

By [9] we can fix [T.sub.1] > 0 such that [mu] is totally ergodic for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (i.e. [mu] is ergodic for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Additionally, it follows from the definitions that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These, Lemma 2 and [mu]([M.sup.*]) = 1 imply that there is K > 0 such that log K [less than or equal to] log [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [mu]-a.e. x [member of] X. From this we obtain log [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, let A = {[A.sub.n]: n [member of] [N.sup.+]} be the sequence of linear maps [A.sub.n]: N [right arrow] N defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever x [member of] [M.sup.*]. Since [M.sup.*] is open and X of class [C.sup.1], we have that A is measurable. Since log [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the Furstenberg-Kesten Theorem implies that there is [lambda] [member of] R such that

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, [lambda] is the upper Lyapunov exponent of A (cf. p. 150 in [12]). Since the Lyapunov exponents {[[bar.[chi]].sub.1], ..., [[bar.[chi]].sub.p]} of A are all negative, we get [lambda] < 0.

Next consider the sequence of functions [f.sub.n]: M [right arrow] R given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [M.sup.*] is open and X of class [C.sup.1], [f.sub.n] is a sequence of measurable functions. Moreover, (1) implies [absolute value of ([f.sup.n])] [less than or equal to] 2[lambda] for large n. Since the constant map x [??] 2[lambda] is integrable (because [mu](M) = 1), (1) and the Dominated Convergence Theorem imply

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Pick 0 < [epsilon] < -[lambda] so [lambda] + [epsilon] < 0. The above limit implies that there is n [member of] [N.sup.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As [mu] is totally ergodic for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is ergodic for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, we are done by taking [T.sub.0] = n[T.sub.1].

In what follows we will denote by B(x, [delta]) and B[x, [delta]] the open and closed balls of radius [delta] of M centered at x [member of] M respectively.

Recall that the support of a Borel probability measure [mu] is the set supp([mu]) of points x [member of] M such that [mu](B(x, [delta])) > 0 for every [delta] > 0. For every flow X and x [member of] M we define the omega-limit set as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We say that [LAMBDA] [subset] M is a transitive set of X if [LAMBDA] = [[omega].sub.X](x) for some x [member of] [LAMBDA].

Lemma 3 and a result by Liao [5] imply Theorem 1 when the involved measure has no singularities in its support. More precisely, we have the following result.

Corollary 4. Let [mu] be a nonatomic ergodic measure of a flow X with supp([mu]) [intersection] Sing(X) = 0. If the Lyapunov exponents of [mu] off the flow direction are all negative, then [mu] is supported on an attracting periodic orbit.

Proof. By Lemma 3 there exists [T.sub.0] > 0 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, by a result of Liao (Lemma 3.2 in [5]) we have that supp([mu]) contains an attracting periodic orbit of X. Since supp([mu]) is ergodic, supp([mu]) is transitive and so [mu] is supported on that periodic orbit. This completes the proof.

2.2. Scaled Poincare flow and [([eta], [T.sub.0]).sup.*]-contractible orbits. Liao defined the scaled linear Poincare flow by

[P.sup.*.sub.t](x) = [parallel]X(x)[parallel]/[parallel]X([X.sub.t](t))[parallel] [P.sub.t](x), [for all]x [member of] [M.sup.*].

By an orbit of X we mean O = {[X.sub.t](x) : t [member of] R}. In such a case we say that O is the orbit through x. The orbit O is regular if X(x) [not equal to] 0.

Given [eta] > 0 and [T.sub.0] > 0 we call a regular orbit O eventually [([eta], [T.sub.0]).sup.*]-contractible if there are x [member of] O and [n.sub.x] [member of] [N.sup.+] such that

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If above [n.sub.x] can be chosen as 1, then O is called [([eta], [T.sub.0]).sup.*]-contractible [7]. Clearly every [([eta], [T.sub.0]).sup.*]-contractible orbit is eventually [([eta], [T.sub.0]).sup.*]-contractible. As in [7], in each case we call x reference point of O.

Given x [member of] M we define [W.sup.sta] (x) as the set of points y [member of] M for which there exists a continuous monotonic function h : [0, [infinity][[right arrow] [0, [infinity][with h(0) = 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Proposition 6.1 of Liao [7] for every flow X and every pair of numbers [eta], [T.sub.0] > 0 there exists [xi] > 0 such that if O is a [([eta], [T.sub.0]).sup.*]-contractible orbit with reference point x, then B(x, [xi][parallel]X(x)[parallel]) [subset] [W.sup.sto](x). The proof is based on the following statistical property (see (1.3) in p. 3 of [8]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since this statistical property is also true for eventually [([eta], [T.sub.0]).sup.*]-contractible orbits with reference point x (just take a limit superior in (2)), Proposition 6.1 in [7] is also true in the eventual case as well. Specifically, we have the following result.

Lemma 5. For every flow X and every pair of numbers [eta], [T.sub.0] > 0 there exists [xi] > 0 such that if O is an eventually [([eta], [T.sub.0]).sup.*]-contractible orbit with reference point x, then B(x, [xi][parallel]X(x)[parallel]) [subset] [W.sup.sta](x).

We say that x is recurrent if x [member of] [[omega].sub.x](x). Denote by R(X) the set of recurrent points of X. A similar definition holds for diffeomorphisms. The following lemma is a consequence of Lemma 3.

Lemma 6. Let [mu] be a nonatomic ergodic measure of a flow X. If the Lyapunov exponents of [mu] off the flow direction are all negative, then there are [eta], [T.sub.0] > 0 and an orbit O which is eventually [([eta], [T.sub.0]).sup.*]-contractible with reference point x [member of] R(X) [intersection] supp([mu]).

Proof. Applying Lemma 3 there are [[eta].sub.0], [T.sub.0] > 0 such that [mu] is an ergodic measure of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Birkhoff's ergodic theorem implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[mu]-a.e. x [member of] M, where [eta] = [[eta].sub.0]/[T.sub.0]. However,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any x [member of] [M.sup.*]. So, the previous inequality yields the following one:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[mu]-a.e. x [member of] M. By Poincare recurrence there is x [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] supp([mu]) satisfying the latter inequality. From this it follows that the orbit O through x is eventually [([eta], [T.sub.0]).sup.*]-contractible with reference point x. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get x [member of] R(X) [intersection] supp([mu]) and we are done.

2.3. Proof of Theorem 1. Let [mu] be a nonatomic ergodic measure of an n-dimensional flow X with n [greater than or equal to] 3. Suppose that the Lyapunov exponents off the flow direction of [mu] are all negative.

Then, by Lemma 6, there are [eta], [T.sub.0] > 0 and an orbit O which is eventually [([eta], [T.sub.0]).sup.*]-contractible with reference point x [member of] R(X) [intersection] supp([mu]). By putting such [eta] and [T.sub.0] in Lemma 5 we obtain [xi] > 0 such that B(x, [xi][parallel]X(x)[parallel]) [subset] [W.sup.sta](x). Taking 2[delta] = [xi][parallel]X(x)[parallel] we get [delta] > 0 satisfying B(x, 2[delta]) [subset] [W.sup.sta](x).

It follows that for every y [member of] B[x, [delta]] there is a continuous monotonic function [h.sub.y] : [0, [infinity][[right arrow] [0, [infinity][ with [n.sub.y](0) = 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since x [member of] R(X), we have x [member of] [[omega].sub.X](x) and so there is a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, by replacing t = [t.sub.k] in the previous limit we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every y [member of] B[x, [delta]]. It follows that for every y [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every z [member of] B(y, [[delta].sub.y]). Since [h.sub.y] is monotonic and B[x, [delta]] has no singularities, we can assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Notice that {B(y, [[delta].sub.y]) : y [member of] B[x, [delta]]} is an open covering of B[x, [delta]]. Since B[x, [delta]] is compact, there are finitely many points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an open covering of B[x, [delta]]. Take z [member of] B[x, [delta]]. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some i = 1, ..., l so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](z) [member of] B[x, [delta]/2]. Hence, the numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are all positive satisfying

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[for all]s [greater than or equal to] 0, we obtain that K is positively invariant, i.e., [X.sub.s](K) [subset] K for every s [greater than or equal to] 0.

We also have that B(x, [delta]/2) [subset] Int(K) and so x [member of] Int(K). Since x [member of] supp([mu]) we conclude that supp([mu]) [intersection] Int(K) [not equal to] 0.

We claim that

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Indeed, take z [member of] K. Hence there are t [greater than or equal to] 0 and y [member of] B[x, [delta]] such that z = [X.sub.t](y). Since y [member of] B[x, [delta]], (3) implies that there is a sequence [i.sub.j] [member of] {1, ..., l} such that

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since each [t.sub.i] > 0, there is r [member of] [N.sup.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By taking this r in (5) we get [X.sub.t](y) = [X.sub.s]([bar.y]) where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since the reversed inclusion is obvious, we obtain (4).

It follows from (4) that K is compact. Since [mu] is ergodic, supp([mu]) is a transitive set. Since supp([mu]) [intersection] Int(K) [not equal to] 0, the positive orbit of q eventually meets Int(K). Since K is compact and positively invariant, supp([mu]) [subset] K.

On the other hand, it is easy to see that Sing(X) [intersection] K = 0 (otherwise there would be some singularities in B[x, [delta]] contradicting that x is regular). Since supp([mu]) [subset] K, we obtain supp([mu])[intersection] Sing(X) = 0. Then, [mu] is supported on an attracting periodic orbit by Corollary 4. This completes the proof.

2010 Mathematics Subject Classification. Primary 37D25; Secondary 37C40.

doi: 10.3792/pjaa.92.131

Acknowledgements. The authors would like to thank Prof. B. Santiago for helpful conversations. The second author was partially supported by CNPq-Brazil and MATHAMSUB 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

References

[1] V. Araujo, Nonzero Lyapunov exponents, no sign changes and Axiom A. (Preprint).

[2] M. Bessa and J. Rocha, Contributions to the geometric and ergodic theory of conservative flows, Ergodic Theory Dynam. Systems 33 (2013), no. 6, 1709 1731.

[3] M. Campanino, Two remarks on the computer study of differentiable dynamical systems, Comm. Math. Phys. 74 (1980), no. 1, 15 20.

[4] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (1985), no. 3, part 1, 617 656.

[5] S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math. 164 (2006), no. 2, 279 315.

[6] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.

[7] S. T. Liao, On (^, d)-contractible orbits of vector fields, Systems Sci. Math. Sci. 2 (1989), no. 3, 193 227.

[8] S. T. Liao, An ergodic property theorem for a differential system, Sci. Sinica 16 (1973), 1 24.

[9] C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math. 23 (1971), 115 122.

[10] Y. Shi, S. Gan and L. Wen, On the singular-hyperbolicity of star flows, J. Mod. Dyn. 8 (2014), no. 2, 191 219.

[11] S. N. Simic, Oseledets regularity functions for Anosov flows, Comm. Math. Phys. 305 (2011), no. 1, 1 21.

[12] M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems 5 (1985), no. 1, 145 161.

Andres Mauricio BARRAGAN * and Carlos Arnoldo MORALES **

(Communicated by Kenji FUKAYA, M.J.A., Nov. 14, 2016)

* Instituto de Ciencias Exatas (ICE), Universidade Federal Rural do Rio de Janeiro, 23890-000 Seropedica, Brazil.

** Instituto de Matematica, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil.
COPYRIGHT 2016 The Japan Academy
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2016 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Barragan, Andres Mauricio; Morales, Carlos Arnoldo
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
Article Type:Report
Geographic Code:1USA
Date:Dec 1, 2016
Words:3810
Previous Article:Hardy's inequality on hardy spaces.
Next Article:Note on a general complex Monge-Ampere equation on pseudoconvex domains of infinite type.
Topics:

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters |