/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

We prove the Liv sic Theorem for arbitrary GL( m;R) cocycles. We con- sider a hyperbolic dynamical system f : X! X and a Holder continuous function A : X! GL(m;R). We show that if A has trivial periodic data, i.e. A(f n 1 p) A(fp)A(p) = Id for each periodic point p = f n p, then there exists a Holder continuous function C : X ! GL(m;R) satisfying A(x) = C(fx)C(x) 1 for all x2 X. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Holder function A.

https://annals.math.princeton.edu/wp-content/uploads/annals-v173-n2-p11-s.pdf

Livšic Theorem for matrix cocycles

This paper discusses relationships among the basic notions
that have been important in recent investigations of
the ergodicity of volume-preserving
partially hyperbolic diffeomorphisms. In particular we survey the possible definitions
of dynamical coherence and discuss the relationship between dynamical coherence and center bunching.

/pdf/dynamical-coherence-and-center-bunching-34u1ffw9n5.pdf

Dynamical coherence and center bunching

We prove the Livsic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a Holder continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there exists a Holder continuous function $C: X \to GL(m,\mathbb R)$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Holder function $A$.

Livsic theorem for matrix cocycles

Hyperbolic Dynamical Systems

In this paper we obtain results about the regularity of the transfer map between two cocycles over an Anosov system, with values in either a diffeomorphism or a Lie group. We also explain how certain examples of de la Llave show that our results are essentially optimal.

/pdf/regularity-of-the-transfer-map-for-cohomologous-cocycles-12rnlud0pb.pdf

Regularity of the transfer map for cohomologous cocycles

We show that among Cr extensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋn of dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

/pdf/transitivity-of-heisenberg-group-extensions-of-hyperbolic-1wcn6pywtw.pdf

Viorel Niţică

Papers

Regularity of the transfer map for cohomologous cocycles

Transitivity of Heisenberg group extensions of hyperbolic systems