UNIVERSIDADE ESTADUAL DE CAMPINAS
SISTEMA DE BIBLIOTECAS DA UNICAMP
REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP
Versão do arquivo anexado / Version of attached file:
Versão do Editor / Published Version
Mais informações no site da editora / Further information on publisher's website:
https://www.tandfonline.com/doi/full/10.1080/14689360802166824
DOI: 10.1080/14689360802166824
Direitos autorais / Publisher's copyright statement:
©2008
by Taylor & Francis. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
CEP 13083-970 – Campinas SP Fone: (19) 3521-6493 http://www.repositorio.unicamp.br
Vol. 23, No. 4, December 2008, 425–435
Relative rotation number for stochastic systems:
dynamical and topological applications
Pedro J. Catuogno, Diego Sebastian Ledesma and Paulo R.C. Ruffino*
Departamento de Matema´tica, Universidade Estadual de Campinas, Campinas, SP, Brazil (Received 7 August 2007; final version received 12 April 2008)
We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.
Keywords: stochastic systems; rotation number; relative rotation; invariance; conjugacy; homotopy
AMS 2000 Subject Classifications: 37E45; 93E03; 37C40 (341C40)
1. Introduction
Many times in physical rotating systems, one may be interested in the relative rotation, in the sense that one considers certain rotation with respect to a referential which can be rotating by itself. In this case, the asymptotic angular behaviour must be compensated by the rotation of the reference. Think, e.g. of the electro-magnetic phenomenon of rotating a bobbin wound inductor in a magnetic field (the external reference) which is also changing directions: the rotation of the magnetic field must be considered as well. In this work we study this situation of rotation with respect to a reference direction which here will be given by a non-vanishing vector X travelling along the trajectories of the system.
Let M be an orientable, boundaryless and connected Riemannian manifold endowed with the corresponding Levi-Civita connection r. We shall consider that M is a two-dimensional manifold, unless otherwise stated (as in Section 3.1). Consider on M a stochastic solution flow ’tof a Stratonovich stochastic differential equation:
dxt¼
Xm i¼0
AiðxtÞ dBit ð1Þ
where Ai, i ¼ 0,1, . . . m are smooth vector fields with bounded derivatives, ðB1
t, B2t, . . . , Bmt Þ
is a Brownian motion in Rmand dB0
t ¼dt. We shall assume that there exists an invariant
probability measures, hence, by locally compactness, there exists an ergodic measure on Mfor this system.
*Corresponding author. Email: [email protected]
ISSN 1468–9367 print/ISSN 1468–9375 online 2008 Taylor & Francis
DOI: 10.1080/14689360802166824 http://www.informaworld.com
Given a curve : Rþ!M, consider two non-vanishing vectors Xt, Yt2T(t)M in the tangent space along . The direction of the vector X will be called the referential with respect to which one calculates angles. The concept of relative rotation number (RRN, for short) which we introduce in this article refers to the asymptotic angular behaviour of Y with respect to X along the curve . In our case will be the trajectories xt¼’t(x0) of the stochastic dynamical systems of Equation (1), where x0 is the initial condition.
The generality of the definition of this RRN turns interesting since many different concepts of rotation number (RN) in literature can be reduced to this context, depending on the choice of the pair X and Y. Among relevant concepts of RNs for non-linear continuous systems, consider the geometrical RN as described in Arnold [1], see also [2], [3] and [4] (in fact introduced by Arnold and San Martin [5]): Let v0be an element in the tangent space Tx0M. Along the trajectories xtof the system there are two other trajectories of vectors in the tangent bundle TM starting at v0 which carries many geometrical and dynamical properties: the linearized flow D’t(v0) and the parallel transport //t(v0). The RN in this context corresponds to the asymptotic angular velocity of Y ¼ D’t(v0) with respect to X ¼ //t(v0). Among other interesting features, this RN measures the asymptotic rotation of the stable and unstable submanifold around the trajectory of a given point, see, e.g. Ruffino [3]. On the other hand, one of the limitations of this approach is the dependence of the RN on the connection r on M, hence, it turns difficult to extract topological properties of the manifold M from RNs. This limitation can be, in some extend, overcome in our RRN proposed here by choosing appropriately X and Y (see Remark 2.2 on asymptotic homology).
These notes are organized in the following way. In Section 2 we introduce the definition of RRN and prove an ergodic theorem of existence a.s. We show how some classical examples of RN are particular case of our RRN. In Section 3 we discuss general properties: (1) In higher dimension, the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may degenerate); (2) We compare the RRN with the geometry of curves in the plane; (3) Robustness is discussed and finally we present an invariance of RRN by conjugacy and by homotopy.
2. Relative rotation number
In this section, we consider M to be a two-dimensional Riemannian manifold, whereas higher dimensional case will be treated in the next section. Given a continuous curve : [0, 1) ! M and a pair of non-vanishing continuous vector processes Xtand Ytliving in the punctured tangent bundle TM0 such that Xt, Yt2T(t)M, we shall denote by A,X,Y(t) the continuous real process which measures the angle between Yt and Xt (called the travelling reference), with initial angle AX,Y(0) 2 [0, 2). We shall define the RRN of the vector Yt, with respect to Xtalong the curve (t) by:
ð, X, Y Þ ¼ lim
t!1
A,X,YðtÞ
t ,
when the limit exists. It measures the asymptotic relative angular velocity of Ytwhen Xt is considered as the angular reference in the plane T(t)M. Note that, although in some calculations normalization will be convenient, the norms of X and Y are not relevant.
Let : TM0!Mbe the projection. Consider in TM0the angular 1-form ! in each punctured tangent space TxM0, x 2 M, see [6]. Let er be the natural vertical extension of
the connection r in M to TM0, i.e.: the er-covariant derivative of a vector field Wtalong a curve Zt2TM0 is given by erZ0W ¼ r
ðZÞ0W. We shall denote also by er for the same
natural extension of r into the sphere bundle STM (the context will make clear when eris in TM0or in STM). Since parallel transport preserves angles, we have that ! is a vertical tensor in TM0, i.e. er! ¼0. In terms of !, the definition of RN above can be restated as:
ð, X, Y Þ ¼ lim t!1 1 t Z Xt ! Z Yt ! ð2Þ With this definition, properties below follow immediately:
Proposition 2.1: Let be a curve in M. Consider X, Y, Z continuous processes in TM0, defined along . Assuming that the RRN involved exists, we have the following relations:
(a) (additivity) (, X, Z ) ¼ (, X, Y ) þ (, Y, Z ); (b) (anti-symmetry) (, X, Y ) ¼ (, Y, X ).
An easy but illustrative example of the dependence of (, X, Y ) on the trajectory of can be seen in the following situation: consider the classical example of X a constant vector field in R2and Y the linear vector field which induces unitary rotation in the plane. If does not turn around the origin infinitely many times then the RN (, X, Y ) is zero. Dependence on the vectors X and Y will become clear with the examples in Section 2.2.
2.1. Existence of the limit
We are interested in the case when the curve is given by trajectories of a stochastic dynamical system, i.e. is the solution xtof the stochastic differential Equation (1) and the tangent processes X and Y are diffusions in TM0 along xt. Without loss of generality, frequently it will be convenient to assume that X and Y live in the unitary sphere bundle STM. They will be described by covariant equation in STM of the following type:
DYt ¼
Xm i¼0
iðYtÞ dWti, ð3Þ
with Y02Tx0M. Analogous equation for Xt2STM.
We prove initially the existence of the RN (xt, X, Y ) in the particular case when X is the parallel transport of Y(x0) 2 Tx0, denoted by X ¼ //Y0.
Consider the auxiliary unitary vector Zt2STxtM orthogonal to Yt such that the
orientation of (Yt, Zt) is positive for each t 0. Using the fact that the angular form ! is parallel, we have that the infinitesimal angle dAðtÞ ¼ !YtðDY Þ ¼ hDYt, Zi. Hence,
the RN can be calculated integrating the inner product: ðxt, ==Y0, Y Þ ¼ lim t!1 1 t Zt 0 DYs, Zs h i:
Our first existence results is the following:
Proposition 2.2: Consider a trajectory xt(!) of the SDS (1) and a diffusion Yt2STM along xtdescribed by Equation(3). Let be an ergodic invariant probability measure in M and be
an invariant measure in STM with ¼ . Then, the RRN (xt, //Y0, Yt) exists and is constant for P a. Every (!, x0). Precisely, P a.s.:
ðxt, ==Y0, Y Þ ¼ Z STM 1 2 Xm i¼1 5 eriðiðvÞÞ, erRiðRiðvÞÞ4þ5 0ðvÞ, Rv 4 d ðvÞ: Proof: Let R denote the rotation on a tangent space TxMby /2 in the positive direction. That is, Zt¼RYtand Ztsatisfies the covariant equation
DZt¼
Xm i¼0
RiðR1ZtÞ dWti:
Hence, from Equation (3) we have the following Itoˆ representation of the infinitesimal rotation: hDYt, Zti ¼ Xm i¼1 5 iðYtÞ, Zt4 dWti þ1 2 Xm i¼1 5 eriðiðYtÞÞ, erRiðRiðYtÞÞ4 dt þ5 0ðYtÞ, Zt4 dt: ð4Þ
By boundedness of the integrand with respect to the martingale, we have
ðxt, ==Y0, Y Þ ¼ lim t!1 1 t Zt 0 1 2 Xm i¼1
5 eriðiðYtÞÞ, erRiðRiðYtÞÞ4 þ 5 0ðYtÞ, Zt4
( )
dt: The ergodic theorem for diffusions states that, if is an ergodic invariant probability measure on STM for the stochastic flow generated by equation (3), then the RN (xt, //Y0, Yt) exists and the formula of the statement holds. œ Note that, unless the RN is zero (e.g. Y is asymptotically the parallel transport //Y0), for each ergodic measure in M, it corresponds to a single invariant measure in STM with ¼ .
Since the projection of non-unitary diffusions in TM0into the unitary sphere bundle STM is again a diffusion, Proposition 2.2 also guarantees that the RN exists for any diffusion Y in TM0. In the general case:
Theorem 2.3: Let xt be a diffusion in M and consider Xt, Yt tangent diffusions in TxtM along xt. If there exists an ergodic invariant measure in M then there exists
(xt, Xt, Yt), the RRN of Y with respect to the referential X along xt, moreover it is P a.s. constant.
Proof: We have by additivity and anti-symmetry of the RRN (Proposition 2.1) that: ðxt, Xt, YtÞ ¼ðxt, ==Y0, YtÞ ðxt, X0, XtÞ,
and both asymptotic parameters on the right side exist P a.s. according to last
Remark 2.1: The concept of RRN introduced here in the context of stochastic systems can be easily interpreted in the context of control systems. This is particularly useful, for example, to perform numerical simulations, or to apply arguments which demand differentiability by parts of trajectories (see, e.g. Theorem 3.1). Consider that both xtand the diffusion Y 2 TM0are driven by control systems:
dxt¼
Xm i¼0
uiðtÞAiðxtÞ ð5Þ
with control function u(t) ¼ (u0(t), . . . , um(t)) measurable and locally square integrable. Analogously:
DYt¼
Xm i¼0
uiðtÞiðYtÞ: ð6Þ
In this context, if one performs the Wong–Zakai approximation where the control functions are polygonal approximations of Brownian motion, one has that the RRN of the control system converges in probability to the RRN of the associated stochastic system.
2.2. Examples
Many classical RNs along continuous dynamical systems in literature can be described in the context of this article by appropriate choices of the pair Ytand the referential vector Xt. We shall present some of these examples. At first we illustrate with three cases the vector fields in STM which appear in the equation of the tangent vectors X and Y in (3). They correspond to some frequently studied tangent processes along continuous dynamical systems as in Equation (1).
Case 1: Y is the parallel transport along xt, i.e. Yt¼//t(Y0), for some initial vector Y02Tx0M. Then the vector fields in the Equation (3) of Ytare
i 0.
Case 2: Yis the linearized flow of the SDS in TM, then its covariant equation in TM0is: DYt¼
Xm i¼0
rAiðYtÞ dBit,
with initial condition Y02Tx0M. Hence
i
(Y ) ¼ rAi(Y ) is linear. In the sphere bundle STM:
DYt ¼
Xm i¼0
rAiðYtÞ hrAiðYtÞ, Y iY
dBit:
Hence i(Y ) ¼ (rAi(Y ) hrAi(Y ), YiY ).
Case 3: Yis a fixed differentiable vector field on the manifold, i.e. Yt¼Y(xt), then, DYt¼ rY dxt ¼X m i¼0 rYðAiÞðx tÞ dBit:
Hence i(Y ) ¼ rY(Ai((Y ))). If the Euler characteristic of M is not zero, one has to check that trajectories does not cross singularities with probability one. Stochastic non-degeneracy of the system guarantees this condition if the set of singularities is finite. Example 1a: The most simple classical concept of RN of trajectories xtwinding around the origin of R2can be described in the following way: we identify the tangent space at any point with R2itself; then this RN corresponds to (0, X0, xt), where X0 is any constant vector (Case 1 above).
Example 1b: For a deterministic non-linear system _x ¼ A(x) in R2, the RN which counts the rotation of the tangent vectors along the curve, e.g. the RN which detects asymptotic spiralling around points which are not necessarily the origin is the following: take (xt, X0, A(xt)); where X0 is any constant vector (Case 1, again) and Y can be described by Case 3.
Example 2: Generalizing Example 1b, consider the RN proposed by Arnold and San Martin [5] for systems in manifolds of dimension higher than two, lately studied for stochastic systems by Ruffino [3], Arnold and Imkeller [2]. For the SDS of Equation (1), this RN corresponds to the RRN (xt, Xt, Yt), where Xt is the parallel transport (Case 1) and Yt is the linearized system (Case 2). This concept of RN is particularly interesting because in dimension two, it measures the asymptotic rotation of the stable or unstable submanifolds, moreover, it does not depend on the choice of the initial vectors X0 and Y0, cf [3].
Example 3: For rotation with respect to an external referential, say, a static magnetic field X, the RRN is given by (xt, X(xt), Yt), where Xtis as described in Case 3 and Ytis as described in Case 2.
Generically speaking, choosing conveniently the referential vector field X, the relative rotation with respect to X can be used to detect entrance or frequency under which trajectories visit certain strategic set. It is not our intention in this notes going further into the analysis of these topological applications. We only mention as remarks:
Remark 2.2 (Asymptotic homology): It is possible to introduce a vector field X in such a way that each passage in a generator of the first homotopy group 1, X rotates once in the sphere bundle TS1Mof a two-dimensional manifold M, cf Schwartzman [7].
Remark 2.3 (Lyapunov exponents calculated as RRN): Given a linear system in R2n{0},
dxt ¼A0xtdt þ
Xm i¼1
AiðxtÞ dBit ð7Þ
where Aj’s are 2 2 matrices. Consider the vector field X(x) ¼ (cos log kxk, sin log kxk). Then, the continuous angle tof X(xt) with respect to the parallel transport //tXis given by log kxtk, hence ðxt, ==X, XðxÞÞ ¼ lim t!1 t t ¼ lim t!1 log kxtk t ¼ ðx0Þ
where (x0) is the Lyapunov exponent of x0, whose existence is guaranteed by the multiplicative ergodic theorem.
3. General properties
In this section we present some general properties of the RRN.
3.1. Higher dimensional systems
Let p be a 2-plane in Rn, n 2 with the corresponding orthogonal projection Pp: R n
!p. Let Zt be a stochastic process in Rn. The projection of Z in p will be appropriate to measure angles in the plane p, at any time t 0, only if it satisfies the following non-vanishing projectionhypothesis:
Hypothesis (NVP): We shall say that a process Z in Rn satisfies the non-vanishing projection (NVP) hypothesis if
PZt2ker Pp, for some t 4 0¼0: Consider two processes Xtand Ytin R
n
which satisfies the NVP. Then, we shall call the relative rotation of Y around X with respect to the plane Pp, the RRN of the corresponding projections (0, PpX, PpY). In non-linear systems in a manifold M, for any purposes, interesting planes are those given by parallel transport //t( p) along the trajectories, where p is a fixed plane in Tx0M.
We shall explain how Hypothesis NVP is only partially restrictive for deterministic systems (Theorem 3.1); and for stochastic systems, it not restrictive at all: if X and Y are non-degenerate diffusions in STM, hence NVP condition is satisfied for any plane p.
Initially, we establish the result for differentiable trajectories as an application of the Sard’s Theorem.
Theorem 3.1: Let t: I R ! Rnþ1 n41 be a differentiable curve. If ktk 6¼0 for all t, then there exists a vector v 2 Sn such that Pv?t6¼0 for all t, where v?is the hyperplane
orthogonal to v.
Proof: Suppose that Pv?t crosses the 0 for all v 2 Sn, i.e. t crosses [v] ¼ { v/ 2 R}.
It follows that the curve
t ¼
t
ktk
is a differential curve that covers an open subset of Sn. So, by Sard’s theorem [8], the image Im( t) must contain regular points. But, if y is a regular value, then y cannot be in the image Im( t) since this fact implies that there is a t0such that t0¼y, with D t0surjective,
which is a contradiction. œ
Note that, in deterministic case, for a fixed curve X 2 Rn, the planes which does not satisfy hypothesis NVP are those in the subsetSt40X?
t of the Grasmannian Grn1(n). When X and Y are diffusions processes, the NVP-hypothesis can be achieved by approximating the trajectories of the diffusions Xtand Ytby trajectories of the associated control system when it generates differentiable by parts trajectories (according to Remark 2.1), then the one may apply Theorem 3.1 to guarantee that the possibilities of choices of p full Lebesgue measure in Gr2(n).
We can refine the conclusion of the paragraph above by exploring probabilistic properties of these processes:
Theorem 3.2: Let X and Y be stochastically non-degenerate diffusions in Rn, n 3, then, hypothesis NVP holds for X and Y for all planes p 2 Gr2(n).
Proof: Consider a stochastically non-degenerate diffusion X in Rn. For a given 2-plane p, there exists an orthonormal basis in Rn such that the process is described by ðX1
t, X2t, . . . , Xdt Þand the projection Ppis described by the first two coordinates. We have to prove that
P X1 t, X2t ¼ ð0, 0Þ for some t 4 0 ¼0: In fact, suppose that at t ¼ 0 we have ððX1
0, X20ÞÞ 6¼ ð0, 0Þ. Then, let n,nþ1¼ !: there exists t 2 ½n, n þ 1Þ; ðX1t, X 2 tÞð!Þ ¼ ð0, 0Þ : The probability that ðX1
t, X2tÞ enters the ball B(0, ) with t in [0, 1] is greater than the
probability that the trajectories hit the origin, moreover it vanishes when goes to zero. Hence, by the time-homogeneity of Markov processes, we have that P{n, nþ1} ¼ 0. Therefore, considering the zero measure set 1¼S1n¼0n,nþ1, we conclude that the NVP
hypothesis holds.
If the initial condition ðX1
0, X20Þ ¼ ð0, 0Þ, the result also holds since the rotation
is considered with t40, i.e. the point t ¼ 0 can be ignored in the integration. To satisfies the NVP hypothesis for X and Y simultaneously, just exclude the union of two
zero-measure sets. œ
3.2. Relative rotation number and the planar geometry
In this section we are back to two-dimensional systems. We compare RRN with the geometry of curves in two-dimensional manifolds.
Proposition 3.3: Let tbe a regular curve on a surface M R 3
and let X be a vector field in M. We shall write Xt¼X(t), then
ðt, ==tðX0Þ, XtÞ ¼ lim T!1 1 T ZT 0 D dt Xt jXtj dt, where ½ðD=dtÞðXt=jXtjÞ is the algebraic value of the covariant derivative.
Proof: Take a positive orthonormal basis of R3 along the curve t: {V1(t), V2(t), N} where V1(t), V2ðtÞ 2 TtMare parallel vectors and N is the normal vector to the surface.
Then we can write
Xt ¼xtV1ðtÞ þ ytV2ðtÞ:
Let Wt ¼ ðXt=jXtjÞ be the normalization of Xt. Then the vector product N ^ Wt¼
ytV1ðtÞ þ xtV2ðtÞ
x2 t þy2t
By definition: DWt dt ¼ 5W0 t, N ^ Wt4:
By parallel transporting back, we get
t ¼==1t ðXtÞ ¼xtV1ð0Þ þ ytV2ð0Þ:
Recalling that the covariant derivatives of V1(t) and V2(t) are zero and that all the calculations are performed in the same tangent space T0M, one finds:
DWt dt ¼xty_tytx_t x2 t þy2t ¼! tð _ tÞ:
Hence the formula holds. œ
Corollary 3.4: Let tbe a regular arc length parameterized curve on a surface M R3then ðt, ==tðv0Þ, t0Þ ¼T!1lim 1 T ZT 0 kgðsÞds,
where kg(s) is the geodesic curvature of t.
Proof: The proof follows from the last proposition and the fact kgðsÞ ¼ ½ðDs0=dÞs . œ
It is well-known that for flat, regular, closed and simple curves, the rotation index is I ¼ 1, depending on the orientation (see, e.g. do Carmo [9]) so that the RN of these curves is simply
ðtÞ ¼
2 T ,
where T is the period of s. Here (t) abbreviates (t, //t(v0), t0) for any non-vanishing v02T(0)M.
Corollary 3.5: Let s be a closed regular arc length parameterized curve in R2 and period T. Then ðtÞ ¼wI, where w ¼2/T and I ¼ 1 2 Z T 0 kðsÞds, is the rotation index of s|(0,T ).
Proof: By the last proposition we see that ðtÞ ¼ lim S!1 1 S ZS 0 kðsÞds: Because of the periodicity of the curve we can make
ðtÞ ¼ lim n!1 1 nT n ZT 0 kðsÞds ¼2 T I: œ
3.3. Discussion on the robustness
From the examples we have presented, one notes that in general the RRN (, X, Y ) is quite sensitive on the trajectory , the referential vector X and on Y. Depending on the description of X and Y (Example 2, above), the RN will depend also on the connection r on the manifold. These sensitiveness on the objects involved can be explored to turn this RN into a relevant dynamical information (Remarks 2.2. and 2.3 illustrate this fact).
No matter the sensitiveness of this parameter, its worth mentioning few situations under which RRN is preserved. Starting with the most obvious: Consider the RRN (, X, Y ). If the referential X is perturbed to another referential eXsuch that their RRN vanishes, i.e. (, X, eXÞ ¼0, then the original RRN is not affected, i.e. ð, eX, Y Þ ¼ ð, X, Y Þ. This is a direct consequence of the additivity stated in Proposition 2.1.
Different from most RNs for continuous dynamical systems proposed in literature (see e.g. Arnold [1], Arnold and Imkeller [2], Stender [10] Ruffino [3] and references therein), the RRN is invariant by conjugacy of dynamical systems in the following sense:
Theorem 3.6 (invariance by conjugacy): Let M and N be orientable two-dimensional manifolds and : M ! N a diffeomorphism. Then, given a curve (or a dynamical system) in M, Xtand Ytvectors in TM
0
along t. If preserves orientation then: ð, X, Y Þ ¼ ððÞ, X, Y ÞÞ:
Otherwise,
ð, X, Y Þ ¼ ððÞ, X, Y ÞÞ:
The proof is trivial since the linear map does not change the asymptotic frequency in
which Ytwinds around and cross the reference Xtin the plane TtM.
Finally we discuss a situation where RRN is invariant by homotopy. The referential X is a fixed vector field in M, a two-dimensional orientable manifold. Y is parallel transport along homotopic curves 0, 1: R
þ
!Mwith 0(0) ¼ 1(0) ¼ p
Theorem 3.7 (Invariance by homotopy): Let H(s, t) : Rþ[0, 1] ! M be a differentiable homotopy between 0 and 1. Let X be a vector field M and write Xts for X(H(s, t)).
Assume that
p, == 1t Xt0, ==1t Xts¼0 for all s 2[0, 1]. Then, for Ys
t ¼==tY0, where Y02TpM and the parallel transport are taken along H(s, ) we have:
0, X0, Y0
¼ Hðs, Þ, Xð s, YsÞ
Proof: Since r! ¼ 0, then the RRN is invariant, when we parallel transport back the vectors X and Y to the initial point H(s, 0) ¼ p. That is, we have, for all s 2 [0, 1]
ðHðs, Þ, Xs, YsÞ ¼ð p, ==1t Xs, ==1t YsÞ ¼ð p, ==1t Xs, Y0Þ:
Using this formula, together with additivity and the antisymmetry of Proposition 2.1: ðHðs, Þ, Xs, YsÞ ð0, X0, Y0Þ ¼ð p, ==t1Xs, Y0Þ ð p, ==1t X0, Y0Þ
¼ð p, ==1t Xs, Y0Þ þð p, Y0, ==1t X0Þ
¼ð p, ==1t Xt0, ==1t XtsÞ:
The result follows since the last term is zero by hypothesis. œ
Acknowledgements
Research by Pedro J. Catuogno was partially supported by FAPESP, grant no. 02/10246-2. Research by Diego Sebastian Ledesma was supported by FAPESP, grant no. 04/13758-0. Research by Paulo R.C. Ruffino was partially supported by CNPq, grant no. 301112/2003-7 and FAPESP, grant no. 02/10246-2.
References
[1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
[2] L. Arnold and P. Imkeller, Rotation numbers for linear stochastic differential equations, Ann. Probab. 27 (1999), pp. 130–149.
[3] P. Ruffino, Rotation numbers for stochastic dynamical systems, Stochastics and Stochastics Reports 60 (1997), pp. 289–318.
[4] P. Ruffino, Decomposition of stochastic flow and rotation matrix, Stochastics and Dynamics 2 (2002), pp. 93–108.
[5] L. Arnold and L. San Martin, A multiplicative ergodic theorem for rotation numbers, J. Dynam. Differ. Equat. 1 (1989), pp. 95–119.
[6] R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Springer, New York, Berlin, 1982.
[7] S. Schwartzman, Asymptotic cycles, Ann. Math. 66 (1957), pp. 270–284.
[8] R. Abraham, J.E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd ed., Springer, New York, Berlin, 1988.
[9] M.P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.
[10] T. Stender, A generalization of imaginary parts of eingenvalues for matrices: chain RNs, Linear Algebra Appl. 426 (2007), pp. 53–70.
