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### Open Access Journals

DCDS

We investigate the open Closing Lemma problem for vector fields on the $2$-dimensional torus. The local $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points.
Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$.

The proof relies on a new result in $1$-dimensional dynamics on the non-existence of semi-wandering intervals of smooth order-preserving circle maps.

The proof relies on a new result in $1$-dimensional dynamics on the non-existence of semi-wandering intervals of smooth order-preserving circle maps.

DCDS

We study the existence of transitive exchange transformations with flips defined on the unit
circle $S^1$. We provide a complete answer to the question of whether there exists a transitive
exchange transformation of $S^1$ defined on $n$ subintervals and having $f$ flips.

DCDS

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I.
Oseledec,

*A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems*, Trudy Moskov. Mat. Obšč.**19**(1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise*splitting*of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a*filtration*of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems**30**(2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed*without*the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.## Year of publication

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