Rigidity of fiber bunched cocycles

We prove a rigidity theorem for fiber bunched matrix-valued Hölder cocycles over hyperbolic homeomorphisms. More precisely, we show that two such cocycles are cohomologous if and only if they have conjugated periodic data.     

Hyperbolic Sequences of Linear Operators and Evolution Maps

For a nonautonomous dynamics defined by a sequence of linear operators on a Banach space, we give a characterization of its nonuniform hyperbolicity in terms of the hyperbolicity of an associated evolution map. We also obtain a generalization of this characterization to the class of nonuniformly hyperbolic cocycles over a locally compact topological space with values on the set of bounded linear operators.     

Cocycles Over Abelian TNS Actions

We study extensions of higher-rank Abelian TNS actions (i.e. hyperbolic and with a special structure of the stable distributions) by compact connected Lie groups. We show that up to a constant, there are only finitely many cohomology classes. We also show the existence of cocycles over higher-rank Abelian TNS actions that are not cohomologous to constant cocycles. This is in contrast to earlier results, showing that real valued cocycles, or small Lie group valued cocycles, over higher-rank Abelian actions are cohomologous to constants.     

Livšic theorems for Banach cocycles: Existence and regularity

We prove a nonuniformly hyperbolic version Liv?ic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure. Moreover, we also prove that a μ-continuous solution of the cohomological equation is actually Hölder continuous for the uniform hyperbolic system, where a map is called μ-continuous if there exists a sequence of compact subsets whose union is of μ-full measure, such that the restriction of the map to each of these compact subsets is continuous.     

Datko-Pazy conditions for nonuniform exponential stability

For linear cocycles over both maps and flows, we obtain Datko-Pazy type of conditions under which all Lyapunov exponents of a given cocycle are negative. Furthermore, by combining our results with the results on subadditive ergodic optimisation, we also present new criteria for uniform exponential stability of linear cocycles.     

Uniform Convergence of Schr?dinger Cocycles over Simple Toeplitz Subshift

For locally constant cocycles defined on an aperiodic subshift, Damanik and Lenz (Duke Math J 133(1): 95–123, 2006) proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. In this paper, we study simple Toeplitz subshifts. We give a criterion that simple Toeplitz subshifts satisfy condition (B), and also give some sufficient conditions that they do not satisfy condition (B). However, we can still prove the uniformity of Schr?dinger cocycles over any simple Toeplitz subshift. As a consequence, the related Schr?dinger operators have Cantor spectrum of Lebesgue measure 0. We also exhibit a fine structure for the spectrum, and this helps us to prove purely singular continuous spectrum for a large class of simple Toeplitz potentials.     

Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows

In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy.     

Direct sums of balanced functions,perfect nonlinear functions,and orthogonal cocycles

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non‐cyclic abelian groups and use it to find all the orthogonal cocycles over Z ₂^t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z ₂^t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008     

A note on stamping

The stamping deformation was defined by Apanasov as the first example of a deformation of the flat conformal structure on a hyperbolic 3-orbifold distinct from bending. We show that in fact the stamping cocycle is equal to the sum of three bending cocycles. We also obtain a more general result, showing that derivatives of geodesic lengths vanish at the base representation under deformations of the flat conformal structure of a finite-volume hyperbolic 3-orbifold.      

Cohomology of permutative cellular automata

We examineU(d) valued cocycles for a ?²⁺ action generated by a mixing, permutative cellular automaton and show that the set of Hölder continuous cocycles (for a given Hölder order) which are cohomologous to constant cocycles is both open and closed in the appropriate topology. A continuous dimension function with values in {0, 1,…,d} is defined on cocycles; a cocycle is cohomologous to a constant precisely when the value isd. Whend=1 (the abelian case) the first (essential) cohomology group is countable. IfU(1)? circle is replaced by a finite subgroup, this cohomology group is finite.     

Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semi-regular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland?s construction of bent functions to additive Hadamard cocycles.     

A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

We use ergodic theory to prove a quantitative version of a theorem of M.A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded.     

Hyperbolic Systems in Gelfand and Shilov Spaces

Ukrainian Mathematical Journal - For systems hyperbolic in Shilov’s sense whose coefficients are continuous functions of time, we study the properties of the Green function in S-type spaces....     

Twisted cocycles of lie algebras and corresponding invariant functions

We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation - so called (α,β,γ)-derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants - we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.     

Twisted product of cocycles and factorization of semi‐regular relative difference sets

In this article, we introduce what we call twisted Kronecker products of cocycles of finite groups and show that the twisted Kronecker product of two cocycles is a Hadamard cocycle if and only if the two cocycles themselves are Hadamard cocycles. This enables us to generalize some known results concerning products and factorizations of central semi‐regular relative difference sets. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 431–441, 2008     

A hyperbolic free boundary problem existence uniqueness and discretization

For a one phase free boundary problem for a linear hyperbolic system with constant coefficients in one space dimension with nonlinear boundary conditions we prove existence, uniqueness and continuous dependence of a Lipschitz continuous solution using the method of characteristics. A semidiscrete version of front tracking is shown to be linearly convergent.     

The Livsic Cocycle Equation for Compact Lie Group Extensions of Hyperbolic Systems

In this article we extend well-known results of Livsic on theregularity of measurable solutions to the cocycle equation.Livsic proved a regularity result for real valued cocycles,but for compact Lie group valued cocycles his result was restrictedto coboundaries (that is, trivial or constant cocycles). Inthis article we prove the complete result. We present several useful applications of these results.     

The giant spiral

Cocycles ofZ ^m actions on compact metric spaces provide a means for constructingR ^m actions or flows, called suspension flows. It is known that allR ^m flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. Whenm=1, examples of cocycles are easy to construct; there is a one-to-one correspondence between cocycles and real valued continuous functions. However, whenm>1 the construction of examples of cocycles becomes more problematic. The only existing class of examples, the close to linear cocycles, have strong linearity properties and are well understood. In fact, when theZ ^m action is uniquely ergodic, all cocycles are close to linear. We will show that in general this need not be the case. We present a method, suggested to us by Hillel Furstenberg, for constructing examples of cocycles whenm>1 and use this method to construct a non close to linear cocycle on a minimalZ ² action.     

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.