scholarly journals -recurrence in skew products

2013 ◽  
Vol 34 (5) ◽  
pp. 1525-1537 ◽  
Author(s):  
JON CHAIKA ◽  
DAVID RALSTON
Keyword(s):  
Growth Rate ◽  
Lyapunov Exponents ◽  
Bounded Variation ◽  
Probability Space ◽  
Irrational Rotation ◽  
Infinite Measure ◽  
Skew Products

AbstractThe rate of recurrence to measurable subsets in a conservative, ergodic infinite-measure-preserving system is quantified by generic divergence or convergence of certain sums given by a function $\omega (n)$. In the context of skew products over transformations of a probability space, we relate this notion to the more frequently studied question of the growth rate of ergodic sums (including Lyapunov exponents). We study in particular skew products over an irrational rotation given by bounded variation $ \mathbb{Z} $-valued functions: first the generic situation is studied and recurrence quantified, and then certain specific skew products over rotations are shown to violate this generic rate of recurrence.

scholarly journals Quantitative Global-Local Mixing for Accessible Skew Products

2021 ◽  
Author(s):  
Paolo Giulietti ◽  
Andy Hammerlindl ◽  
Davide Ravotti
Keyword(s):  
Spectral Measure ◽  
Low Frequency ◽  
Almost Periodic ◽  
Careful Choice ◽  
Rapid Mixing ◽  
Infinite Measure ◽  
Skew Products ◽  
Transfer Operators ◽  
Frequency Behavior ◽  
Dense Set

AbstractWe study global-local mixing for a family of accessible skew products with an exponentially mixing base and non-compact fibers, preserving an infinite measure. For a dense set of almost periodic global observables, we prove rapid mixing, and for a dense set of global observables vanishing at infinity, we prove polynomial mixing. More generally, we relate the speed of mixing to the “low frequency behavior” of the spectral measure associated to our global observables. Our strategy relies on a careful choice of the spaces of observables and on the study of a family of twisted transfer operators.

Positive Lyapunov exponents for a dense set of bounded measurable SL(2, ℝ)-cocycles

1992 ◽  
Vol 12 (2) ◽  
pp. 319-331 ◽  
Author(s):  
Oliver Knill
Keyword(s):  
Lyapunov Exponents ◽  
Probability Space ◽  
Image Position ◽  
Dense Set ◽  
Standard Probability

AbstractLet T be an aperiodic automorphism of a standard probability space (X, m). We prove, that the setis dense in L∞(X, SL(2, ℝ)).

REMARKS ON SKEW PRODUCTS OVER IRRATIONAL ROTATIONS

2005 ◽  
Vol 15 (11) ◽  
pp. 3675-3689 ◽  
Author(s):  
L. M. LERMAN
Keyword(s):  
Sufficient Conditions ◽  
Irrational Rotation ◽  
Invariant Sets ◽  
Asymptotically Stable ◽  
Skew Product ◽  
Almost Periodic ◽  
Invariant Curves ◽  
Skew Products ◽  
Irrational Rotation Number ◽  
Periodic Equation

We prove several results of the orbit behavior of skew product diffeomorphisms generated by quasi-periodic differential systems. The first diffeomorphism is derived from a periodic differential equation on the circle by means of a construction proposed by Z. Opial to get a scalar quasi-periodic equation with all its solutions bounded but without an almost periodic solution. We consider both possible cases for the irrational rotation number, transitive and singular (intransitive). The main result for a transitive case is that the related skew product diffeomorphism has a foliation into invariant curves with pure irrational rotation on each curve (being the same for each curve). For intransitive case, we get invariant sets of two types: a collection of continuous invariant curves and invariant sets being dimensionally inhomogeneous ones.Section 3 is devoted to perturbations of a skew product diffeomorphism over an irrational rotation being initially foliated into invariant curves. We prove an analog of Poincaré–Pontryagin theorem which sets conditions when a perturbation of a one-degree-of-freedom Hamiltonian system (given in an annulus and written down in action-angle variables) has limit cycles. Our theorem provides sufficient conditions when a perturbation of a foliated skew product diffeomorphism has isolated invariant curves (asymptotically stable or unstable).In Sec. 4 we present some results on the geometry of skew product diffeomorphisms derived by a quasi-periodic Riccati equation.

scholarly journals Entropy dimension for deterministic walks in random sceneries

2021 ◽  
pp. 1-18
Author(s):  
DOU DOU ◽  
KYEWON KOH PARK
Keyword(s):  
Growth Rate ◽  
Dynamical Systems ◽  
Base System ◽  
Irrational Rotations ◽  
Intermediate Growth ◽  
Skew Products ◽  
Entropy Dimension ◽  
Deterministic Walks ◽  
Different Levels ◽  
Bernoulli Systems

Abstract Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

Typical properties of periodic Teichmüller geodesics: Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
URSULA HAMENSTÄDT
Keyword(s):  
Growth Rate ◽  
Vector Bundle ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Moduli Space ◽  
Abelian Differentials ◽  
The Matrix ◽  
Teichmüller Flow

Abstract Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents

2008 ◽  
Vol 28 (1) ◽  
pp. 245-266 ◽  
Author(s):  
DANIEL SCHNELLMANN
Keyword(s):  
Lyapunov Exponents ◽  
Quadratic Maps ◽  
Skew Products ◽  
Expanding Map

AbstractWe study an extension of the Viana map where the base dynamics is a discontinuous expanding map, and prove the existence of two positive Lyapunov exponents.

scholarly journals On torus homeomorphisms semiconjugate to irrational rotations

2014 ◽  
Vol 35 (7) ◽  
pp. 2114-2137 ◽  
Author(s):  
T. JÄGER ◽  
A. PASSEGGI
Keyword(s):  
Simple Proof ◽  
Irrational Rotation ◽  
Rotation Vector ◽  
Topological Properties ◽  
Empty Interior ◽  
Irrational Rotations ◽  
Skew Products ◽  
Invariant Foliation ◽  
Special Case

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

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