Franks, J. Homology and Dynamical Systems. CBMS Regional Conference Series. 49 (1982).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....block for Phi, with maximal invariant set J ae N , and there is a hyperbolic splitting along J with stable direction parallel to the s coordinate and unstable direction parallel to the u coordinate in each piece. Also, J is carried by the N . Using symbolic dynamics arguments from [Bow72] or [Fra82], the invariant set J is 1 dimensional. For each rectangle R x = I x , there is a product structure R x J x , where C x ) C x ) are compact and totally disconnected. We have loc (J ) R x = I x and W loc (J ) R x = C x . There is a 1 1 correspondence between directed ....

John Franks, Homology and dynamical systems, CBMS regional conference series, no. 49, Amer. Math. Soc., 1982.

....0. ii) Assume there exists an odd d 1 such that every periodic orbit of period j j = d is untwisted, i.e. Delta( 1. Then if any such has least period p (which therefore divides d) then there are periodic points of prime period 2 k d, k 0. More results of this nature can be found in [Fr1]. More generally, mixtures of dynamical and topological methods to prove the existance of periodic points, are important in recent results as new proofs of the Poincar e Birkhoff theorem [Fr2] and the Klingenberg conjecture on closed geodesics on S 2 [Fr4] However, such results lie beyond the ....

J. Franks, Homology and Dynamical systems, Regional Conf. Series vol.49, Amer. Math. Soc., Providence, 1982.

.... 0. ii) Assume there exists an odd d 1 such that every periodic orbit of period j j = d is untwisted, i.e. 1. Then if any such has least period p (which therefore divides d) then there are periodic points of prime period 2 k d, k 0. More results of this nature can be found in [Fr1]. More generally, mixtures of dynamical and topological methods to prove the existance of periodic points, are important in recent results as new proofs of the Poincar e Birkho theorem [Fr2] and the Klingenberg conjecture on closed geodesics on S 2 [Fr4] However, such results lie beyond the ....

J. Franks, Homology and Dynamical systems, Regional Conf. Series vol.49, Amer. Math. Soc., Providence, 1982.

.... [BowL] that the zeta function is rational if and only if there are integral matrices C; D such that for all n, f n = trC n Gamma trD n : It is sometimes the case in dynamics that one can prove the rationality of the zeta function for interesting systems, precisely by finding such matrices [F1,Fri1,Fri2]. The rationality of the zeta function then sharply and transparently captures this constraint. 4.3 Product formula. The zeta function can be written as a (usually infinite) product, i S (z) Y [1 Gamma z n ] Gamma1 where there is a term [1 Gamma z n ] Gamma1 for each S orbit of ....

J. Franks, Homology and Dynamical Systems, CBMS 49 (1982), A.M.S., Providence, Rhode Island.

....has proven to be an extremely important means of relating the local dynamics about hyperbolic fixed points with the global topology of the underlying manifold. Again, these ideas can be extended into systems with periodic orbits, i.e. MorseSmale diffeomorphisms and flows, and further ([5]) With the discovery of the ubiquity of much more complicated dynamics, often referred to as chaos, and with the realization of the important role it plays in the long term or asymptotic dynamics of nonlinear systems, attempts to characterize this behavior in terms of algebraic invariants have ....

J. Franks, Homology and Dynamical Systems, CBMS Reg. Conf. Ser. in Math., 49, AMS, Providence, 1982.

....of prime period 2 k d, k 0. ii) Assume there exists an odd d 1 such that every periodic orbit of period j j = d has Delta( 1. Then if any such has least period pjd then there are periodic points of prime period 2 k d, k 0. Remark. More results of this nature can be found in [Fr1, Fr2]. More generally, mixtures of dynamical and topological methods to prove the existence of periodic points are important in such recent results as new proofs of the Poincar e Birkhoff theorem and the Klingenberg Conjecture on closed geodesics on S 2 . 3. Proving rationality 3.1. Zeta functions ....

J. Franks, Homology and Dynamical systems, Regional Conf. Series vol.49, Amer. Math. Soc., Providence, 1982.

....inequalities, has proven to be an extremely important means of relating the local dynamics about hyperbolic fixed points with the global topology of the underlying manifold. Again, these ideas can be extended into systems with periodic orbits, i.e. MorseSmale diffeomorphisms and flows, and further [5]. With the discovery of the ubiquity of much more complicated dynamics, often referred to as chaos, and with the realization of the important role it plays in the long term or asymptotic dynamics of nonlinear systems, attempts to characterize this behavior in terms of algebraic invariants have ....

....equality is at the heart of the computations of this section. Proposition 3.1 If #(G) 1, then for all z 2 C, det(I Gamma zA) Y 2G det i I Gamma z p( A( j 1=p( 3) Proof. Using the fact that the cardinality of G is finite to reverse the order of summation and Lemma 5. 2 [5] one has that, Gamma ln(det(I Gamma zA) Gamma ln det I Gamma z l X i=0 A i = 1 X m=1 tr i ( P l i=0 A i ) m j m z m = 1 X m=1 X 2f0; lg m tr(A( m z m = X 2G 1 X n=1 tr(A( n ) np( z np( X 2G 1 p( 1 X n=1 tr(A( n ) ....

J. Franks, Homology and Dynamical Systems, CBMS Reg. Conf. Ser. in Math., 49, AMS, Providence, 1982.

....= 0 the system in Equation (1) satisfies the conditions of Theorem 10 in Cohen (1992) Therefore the system will converge to one of the critical points of V (x) for all initial conditions. The systems in Equation (1) are a non autonomous special case of the gradient like systems presented in Franks (1982). This means that they are not able to represent arbitrary dynamics. Note that if the second term on the right hand side of Equation (1) is replaced by the term P n i=2 Q i (x) rxH i (x) where H i (x) are in general different potential functions, then the resulting system can describe arbitrary ....

Franks, J. (1982). Homology and dynamical systems, Vol. 49 of Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI.

....of the form x = GammaP (x) rxV (x) 2.1) where P (x) is a matrix function which is symmetric (i.e. P y = P ) and positive definite at every point x, and where rxV (x) V x1 ; V x2 ; V xn ] y . These systems are a special case of the Morse gradient flows discussed in Franks (1982). Since the matrix function P (x) is symmetric, it is positive definite if all of the eigenvalues are positive. This matrix defines a way to measure distance, or a Riemannian metric, which may change at each point. Relative to this distance measure, the components of the vector field formed by ....

Franks, J. (1982). Homology and dynamical systems, Vol. 49 of Regional Conference Series in Mathematics.

No context found.

Franks, J. Homology and Dynamical Systems. CBMS Regional Conference Series. 49 (1982).

No context found.

J. Franks, Homology and Dynamical Systems, CBMS, vol. 49, Amer. Math. Soc., 1982.

No context found.

J. Franks, Homology and dynamical systems, Regional Conference series 49, Amer. Math. Soc., Providence, R.I., 1982.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC