On Aubry-Mather sets (original) (raw)

Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

Ergodic Theory and Dynamical Systems, 1988

We prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

Journal of Difference Equations and Applications, 2005

This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)- y, x) of the plane R^2, in which F_{ab}(x)= ax if x>0 and bx if x<0. These maps are area-preserving homeomorphisms of the plane that map rays from the origin into rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This difference equation can be written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which \mu= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper studies the set of parameter values where T_{ab} has at least one nonzero bounded orbit, which corresponds to an l_{\infty} eigenfunction of the difference operator. It shows that the for transcendental \mu the set of allowed energy values E for which there is a bounded orbit is a Cantor set. Numerical simulations suggest that this Cantor set have positive one-dimensional measure for all real values of \mu.

Rotation intervals for chaotic sets

1998

Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

Journal of Difference Equations and Applications, 2005

This paper studies the behavior under iteration of the maps T ab (x, y) = (F ab (x)−y, x) of the plane R 2 , in which F ab (x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the nonlinear difference equation x n+2 = 1/2(a − b)|x n+1 | + 1/2(a + b)x n+1 − x n. This family of piecewise-linear maps has the parameter space (a, b) ∈ R 2. These maps are area-preserving homeomorphisms of R 2 that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map S ab : S 1 → S 1 of the circle, which has a well-defined rotation number. This paper characterizes the possible dynamics under iteration of T ab when the auxiliary map S ab has rational rotation number. It characterizes cases where the map T ab is a periodic map.

Universal scaling of critical quasiperiodic orbits in a class of twist maps

Journal of Physics A: Mathematical and General, 1998

Recently we have shown that the fractal properties of the critical invariant circles of the standard map, as summarized by the f (α) spectrum and the generalized dimensions D(q), depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in (Burić N, Mudrinić M and Todorović K 1997 J. Phys. A: Math. Gen. 30 L161). In the present paper this result is extended on the whole class of sufficiently smooth area-preserving twist maps of cylinders. We present numerical evidence that the f (α) and D(q) are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail.

Entropy of twist interval maps

Israel Journal of Mathematics, 1997

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps.

Persistence of homoclinic orbits for billiards and twist maps

Nonlinearity, 2004

We consider the billiard motion inside a C 2-small perturbation of a ndimensional ellipsoid Q with a unique major axis. The diameter of the ellipsoid Q is a hyperbolic two-periodic trajectory whose stable and unstable invariant manifolds are doubled, so that there is a n-dimensional invariant set W of homoclinic orbits for the unperturbed billiard map. The set W is a stratified set with a complicated structure. For the perturbed billiard map the set W generically breaks down into isolated homoclinic orbits. We provide lower bounds for the number of primary homoclinic orbits of the perturbed billiard which are close to unperturbed homoclinic orbits in certain strata of W. The lower bound for the number of persisting primary homoclinic billiard orbits is deduced from a more general lower bound for exact perturbations of twist maps possessing a manifold of homoclinic orbits.

Invariant sets for discontinuous parabolic area-preserving torus maps

Nonlinearity, 2000

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Nonlinearity, 2013

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.

Reversing Symmetries in a Two-Parameter Family of Area-Preserving Maps

Progress of Theoretical Physics, 2005

The integrable twist map is used to derive an invertible two-parameter family of areapreserving maps specified by two arbitrary functions of a real variable. It has been shown that the integrable case has infinitely many reversing symmetries (not necessarily all involutory). Almost all of these symmetries are destroyed under addition of non-integrable terms. However some involutory reversing symmetries survive if certain restrictions on the functions are imposed. An involutory reversing symmetry exists in three different cases. In the first two cases, the symmetry lines are continuous curves leading to infinitely many symmetric periodic orbits. The third case is an interesting example of an involutory reversing symmetry in which the corresponding symmetry lines are isolated points leading to none or few symmetric periodic orbits. The combined restrictions of the first two cases lead to a family of maps with double reversing symmetries. Furthermore there exists a subset of this family (identified by an additional restriction) with quadruple reversing symmetries.