On the Closing Lemma problem for the torus
Simon Lloyd
Discrete & Continuous Dynamical Systems - A 2009, 25(3): 951-962 doi: 10.3934/dcds.2009.25.951
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.
keywords: Cherry flow Black cell. Wandering interval Denjoy theorem
Transitive circle exchange transformations with flips
Carlos Gutierrez Simon Lloyd Vladislav Medvedev Benito Pires Evgeny Zhuzhoma
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 251-263 doi: 10.3934/dcds.2010.26.251
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.
keywords: orientation reversing. interval exchange transformation Rauzy induction
A semi-invertible Oseledets Theorem with applications to transfer operator cocycles
Gary Froyland Simon Lloyd Anthony Quas
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 3835-3860 doi: 10.3934/dcds.2013.33.3835
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.
keywords: covariant vector Lyapunov vector Multiplicative ergodic theorem equivariant subspace.

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