Homoclinic processes and invariant measures for hyperbolic toral automorphisms

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L₂ with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.     

Invariant measures and asymptotics for some skew products

For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures. These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic ergodic theorems, and bounded rational ergodicity.     

Invariant measures for skew products and uniformly distributed sequences

??Almost all?? sequences (r ₁, . . . , r _n , . . . ) of positive integers have the following ??universal?? property: Whenever (X,???) is a Borel probability compact metric space, and ?? ₁, ?? ₂, . . . , ?? _n , . . . a sequence of commuting measure preserving continuous maps on (X,???), such that the action (by composition) on (X,???) of the semigroup with generators ?? ₁, . . . ,?? _n , . . . is uniquely ergodic and equicontinuous, then for every ${x \in X}$ the sequence w ₁,w ₂, . . . , w _n , . . . where $$w_n:=\varPhi_{r_n}(\varPhi_{r_{n-1}}(\ldots(\varPhi_{r_2}(\varPhi_{r_1}(x)))\ldots))$$ is uniformly distributed for???. This is a contribution to Problem 116 of Schreier and Ulam in the Scottish Book.     

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL. The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac & Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schrödinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

     

Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products

We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents.     

Rotationally invariant hyperbolic waves

We use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves. The equation can include weak dissipative, dispersive, or diffractive effects. We give applications to equations from magnetohydrodynamics, elasticity, and viscoelasticity.     

Disintegration of projective measures

In this paper, we study a class of quasi-invariant measures on paths generated by discrete dynamical systems. Our main result characterizes the subfamily of these measures which admit a certain disintegration. This is a disintegration with respect to a random walk Markov process which is indexed by the starting point of the paths. Our applications include wavelet constructions on Julia sets of rational maps on the Riemann sphere.

     

Horseshoes for diffeomorphisms preserving hyperbolic measures

We give extensions of Katok’s horseshoe constructions, comment on related results, and provide a self-contained proof. We consider either a \(C^{1+\alpha }\) diffeomorphism preserving a hyperbolic measure or a \(C^1\) diffeomorphism preserving a hyperbolic measure whose support admits a dominated splitting.     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Let be a compact manifold which is invariant and normally hyperbolic with respect to a semiflow in a Banach space. Then in an -neighborhood of there exist local center-stable and center-unstable manifolds and , respectively. Here we show that and may each be decomposed into the disjoint union of submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.

     

Equivalent invariant measures

LetG be a finitely-generated group of non-singular measurable transformations of a measure space (X, β,p). FixA∈β withp(A)>0. A general technique for groups gives sufficient conditions for there to exist aG-invariant measure ν equivalent top with ν((A)=1. These conditions are phrased in terms of the growth behavior ofg→p(gB) forB∈β. The question of necessity is handled in some special cases.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

A method for calculating invariant subspaces of symmetric hyperbolic equations

An algorithm is constructed for calculating invariant subspaces of symmetric hyperbolic systems arising in electromagnetic, acoustic, and elasticity problems. Discrete approximations are calculated for subspaces that correspond to minimal eigenvalues and smooth eigenfunctions. Difficulties related to the presence of an infinite-dimensional kernel in the differential operator are successfully handled. The efficiency of the algorithm is demonstrated using acoustics equations.