Skew-product Maps Research Articles

Distribution of cycles for one-dimensional random dynamical systems

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of ‘random cycles’, with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms. We also prove another almost-sure convergence theorem, as well as an averaged (annealed) theorem that is related to semigroup actions. We apply our results to the random β-expansion of real numbers, and obtain almost-sure convergences of average digital quantities in random β-expansions of random cycles that do not follow from the application of the ergodic theorems of Birkhoff or Kakutani. Our main results are applicable to random dynamical systems generated by finitely many maps with common neutral fixed points.

Universal supercritical behavior for some skew-product maps

<p style='text-indent:20px;'>We consider skew-product maps over circle rotations <inline-formula><tex-math id="M1">\begin{document}$ x\mapsto x+\alpha\; $\end{document}</tex-math></inline-formula> with factors that take values in <inline-formula><tex-math id="M2">\begin{document}$ {{{\rm{SL}}}}(2,{\mathbb{R}}) $\end{document}</tex-math></inline-formula>. In numerical experiments with <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> the inverse golden mean, Fibonacci iterates of almost Mathieu maps with rotation number <inline-formula><tex-math id="M4">\begin{document}$ 1/4 $\end{document}</tex-math></inline-formula> and positive Lyapunov exponent exhibit asymptotic scaling behavior. We prove the existence of such asymptotic scaling for "periodic" rotation numbers and for large Lyapunov exponent. The phenomenon is universal, in the sense that it holds for open sets of maps, with the scaling limit being independent of the maps. The set of maps with a given periodic rotation number is a real analytic manifold of codimension <inline-formula><tex-math id="M5">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> in a suitable space of maps.</p>

Asymptotic scaling and universality for skew products with factors in SL(2,)

AbstractWe consider skew-product maps over circle rotations $x\mapsto x+\alpha \;(\mod 1)$ with factors that take values in ${\textrm {SL}}(2,{\mathbb {R}})$ . In numerical experiments, with $\alpha $ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

Golden mean renormalization for the almost Mathieu operator and related skew products

Considering SL(2,R) skew-product maps over circle rotations, we prove that a renormalization transformation associated with the golden mean α* has a nontrivial periodic orbit of length 3. We also present some numerical results, including evidence that this period 3 describes scaling properties of the Hofstadter butterfly near the top of the spectrum at α* and scaling properties of the generalized eigenfunction for this energy.

Distribution of Typical Orbits for Random Dynamics Generated by Finitely Many Rational Maps

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, $$R_{1}, R_{2}, \ldots , R_{N}$$ , each with degree $$d_{i};\ 1 \le i \le N$$ and with at least one $$R_{i}$$ in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanski (equilibrium) measure associated to some real-valued Holder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

Parabolic invariant tori in quasi-periodically forced skew-product maps

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ:Rn×Td→Rn×Td:φ(zθ)=(z+ϕ(z)+h(z,θ)+ϵf(z,θ)θ+ω), where ϕ:Rn→Rn is a homogeneous function of degree l with l≥2 and h=O(|z|l+1). We obtain the following results: (a) For n=1, l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For n=1, and ϵ sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition. (ii) First order average is non-zero and ω satisfies the Brjuno-type weak non-resonant condition; (iii) l=2, first order average is zero, both first and second order non-average parts are small enough and ω satisfying Brjuno-type weak non-resonant condition; (iv) l>2, first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and ω satisfies the Brjuno-type weak non-resonant condition. (c) In the case n>1, if first order average belongs to the interior of the range of ϕ, Spec(Dϕ)∩iR=∅, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for ϵ sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.

On trigonometric skew-products over irrational circle-rotations

<p style='text-indent:20px;'>We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach.</p>

On the Lyapunov Exponents for a Class of Circle Diffeomorphisms Driven by Expanding Circle Endomorphisms

We consider skew-product maps on mathbb {T}^2 of the form F(x,y)=(bx,x+g(y)) where g:mathbb {T}rightarrow mathbb {T} is an orientation-preserving C^2-diffeomorphism and bge 2 is an integer. We show that the fibred (upper and lower) Lyapunov exponent of almost every point (x, y) is as close to int _mathbb {T}log (g'(eta ))deta as we like, provided that b is large enough.

Renormalization and universality of the Hofstadter spectrum

We consider a renormalization transformation for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over circle-rotations by the golden mean, we find several periodic orbits for , and we conjecture that there are infinitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evidence that these values describe (among other things) local scaling properties of the Hofstadter spectrum.

Strong orthogonality between the Möbius function and skew products

For $$\tau >3$$ and $$\alpha \in \mathbb {R}$$, let T be a skew product map of the form $$T(x_1,x_2)=(x_1+\alpha ,x_2+h(x_1))$$ on $$\mathbb {T}^2$$ over a rotation of the circle, for which h is of zero topological degree and of class $$C^{\tau }$$. We prove that for a measure-theoretically generic set of $$\alpha $$, such a $$C^{\tau }$$ skew product map T is strongly orthogonal to the Mobius function. Moreover, we establish the strong orthogonality between the Mobius function and some irregular $$C^{\tau }$$ skew product map on $$\mathbb {T}^2$$. The key tools used are Fourier analysis and exponential sums estimate concerning the Mobius function.

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step Skew-Products and Elliptic Cocycles

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of N, $${N\ge2}$$ , symbols and with C1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $${2\times 2}$$ matrix cocycles and our results apply to an open and dense subset of elliptic $${\mathrm{SL}(2,\mathbb{R})}$$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.

On–off intermittency and chaotic walks

We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

A minimal distal map on the torus with sub-exponential measure complexity

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

Li–Yorke sensitivity does not imply Li–Yorke chaos

We construct an infinite-dimensional compact metric space $X$ , which is a closed subset of $\mathbb{S}\times \mathbb{H}$ , where $\mathbb{S}$ is the unit circle and $\mathbb{H}$ is the Hilbert cube, and a skew-product map $F$ acting on $X$ such that $(X,F)$ is Li–Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li–Yorke sensitivity implies Li–Yorke chaos [Akin and Kolyada. Li–Yorke sensitivity. Nonlinearity 16, (2003), 1421–1433].

Parametrization of unstable manifolds and Fatou disks for parabolic skew-products

We study the existence of Fatou components on parabolic skew product maps. We focus on skew products in which each coordinate has a fixed point that is parabolic. As in the geometrically attracting case, we prove that there exists maps F that have one-dimensional disks that are mapped to a point in the Julia set of the restriction of F to an invariant one-dimensional fiber. We first prove a linearization theorem for a one-dimensional map, then for a parabolic skew product. Finally, we apply this result to construct the skew product map described above.

Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products

We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents.

Growth of number of periodic orbits of one family of skew product maps

ABSTRACTIn this article we introduce a one-parameter family of skew product (Gt)t ∈ [−ε, ε] maps exhibiting a heterodimensional cycle such that the number of isolated periodic orbits inside it has not super-exponential growth. The dynamics in the central direction of the maps Gt is described by a one-parameter family of system of iterated functions.

Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+\epsilon a(x_n,y_n,\epsilon), \quad y_{n+1} = T_\epsilon y_n, \end{align*} where $T_\epsilon$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $\epsilon\to0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = \epsilon a(x,y,\epsilon), \quad \dot y = g_\epsilon(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.

Nonhyperbolic step skew-products: Ergodic approximation

We study transitive step skew-product maps modeled over a complete shift of k, k≥2, symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents.We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves.Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak⁎ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.

Möbius disjointness for analytic skew products

For $\tau>2$, let $T$ be a $C^\tau$ skew product map of the form $(x+\alpha,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the M\"{o}bius sequence. This in particular implies that any non-uniquely ergodic $C^\tau$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the M\"{o}bius sequence.