Skew Products Research Articles

Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on $$\mathbb {T}^4$$

Uniqueness of u-Gibbs Measures for Hyperbolic Skew Products on $$\mathbb {T}^4$$

An Open Set of Skew Products with Invariant Multi-graphs and Bony Multi-graphs

An Open Set of Skew Products with Invariant Multi-graphs and Bony Multi-graphs

Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles

For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimensional central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variational principle for nonhyperbolic (zero exponent) ergodic measures. The result is applied to the setting of \(\mathrm {SL}(2,\mathbb {R})\) matrix cocycles and provides a counterpart to Furstenberg’s classical result: for an open and dense subset of elliptic \(\mathrm {SL}(2,\mathbb {R})\) cocycles we construct ergodic measures with upper Lyapunov exponent zero and with metric entropy arbitrarily close to the topological entropy of the set of infinite matrix products with subexponential growth of the norm.

Non-Uniform Hyperbolicity in Polynomial Skew Products

AbstractLet $f:\mathbb {C}^2\to \mathbb {C}^2$ be a polynomial skew product that leaves invariant an attracting vertical line $ L $. Assume moreover $f$ restricted to $L$ is non-uniformly hyperbolic, in the sense that $f$ restricted to $L$ satisfies one of the following conditions: (1) $f|_L$ satisfies topological Collet–Eckmann and weak regularity conditions. (2) The Lyapunov exponent at every critical value point lying in the Julia set of $f|_{L}$ exists and is positive, and there is no parabolic cycle. Under one of the above conditions we show that the Fatou set in the basin of $L$ coincides with the union of the basins of attracting cycles, and the Julia set in the basin of $L$ has Lebesgue measure zero. As an easy consequence there are no wandering Fatou components in the basin of $L$.

Quantitative Global-Local Mixing for Accessible Skew Products

We 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 Exponents for Random Products of Conservative Surface Diffeomorphisms and Some Skew Products

Positive Exponents for Random Products of Conservative Surface Diffeomorphisms and Some Skew Products

Periodic Behavior of Maps Obtained by Small Perturbations of Smooth Skew Products

Periodic Behavior of Maps Obtained by Small Perturbations of Smooth Skew Products

Möbius Disjointness for Skew Products

Abstract We show that for $\varepsilon> 0$, every $C^{1+\varepsilon }$ skew product on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$ satisfies Sarnak’s conjecture. This is an improvement of earlier results of Kułaga–Przymus–Lemańczyk, Huang–Wang–Ye, and Kanigowski–Lemańczyk–Radziwiłł.

Smale endomorphisms over graph-directed Markov systems

AbstractWe study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

Skew product Smale endomorphisms over countable shifts of finite type

We introduce and study skew product Smale endomorphisms over finitely irreducible shifts with countable alphabets. This case is different from the one with finite alphabets and we develop new methods. In the conformal context we prove that almost all conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional and their dimension is equal to the ratio of (global) entropy and Lyapunov exponent. We show that the exact dimensionality of conditional measures on fibers implies global exact dimensionality of the original measure. We then study equilibrium states for skew products over expanding Markov–Rényi transformations and settle the question of exact dimensionality of such measures. We apply our results to skew products over the continued fraction transformation. This allows us to extend and improve the Doeblin–Lenstra conjecture on Diophantine approximation coefficients to a larger class of measures and irrational numbers.

Non-wandering Fatou Components for Strongly Attracting Polynomial Skew Products

We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with an invariant attracting fiber, under the assumption that the multiplier $ \lambda $ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component.

On a Class of Totally Topologically Transitive Skew Products Defined on Cells in $\mathbb R^n$, ${n\ge 2}$

В настоящей работе получены достаточные условия полной топологической транзитивности (транзитивности всех итераций рассматриваемого отображения) $C^3$-гладких косых произведений одного класса, действующих на клетках в $\mathbb R^n$, $n\ge 2$. Библиография: 16 названий.

Nonwandering Sets of C 1-Smooth Skew Products of Interval Maps with Complicated Dynamics of Quotient Map

We obtain exact formulas describing the nonwandering set of a C 1-smooth skew product of interval maps with Ω-stable quotient map of type ≻ 2∞.

Milnor Attractors of Skew Products with the Fiber a Circle

We prove that for a generic skew product with circle fiber over an Anosov diffeomorphism the Milnor attractor (also called the likely limit set) coincides with the statistical attractor, is Lyapunov stable, and either has zero Lebesgue measure or coincides with the whole phase space. As a consequence we conclude that such skew product is either transitive or has non-wandering set of zero measure. The result is proved under the assumption that the fiber maps preserve the orientation of the circle, and the skew product is partially hyperbolic.

Sternberg Linearization Theorem for Skew Products

We present a special kind of normalization theorem: linearization theorem for skew products. The normal form is a skew product again, with the fiber maps linear. It appears that even in the smooth case, the conjugacy is only Holder continuous with respect to the base. The normalization theorem mentioned above may be applied to perturbations of skew products and to the study of new persistent properties of attractors.

Skew products of interval maps over subshifts

We treat step skew products over transitive subshifts of finite type with interval fibers. The fiber maps are diffeomorphisms on the interval; we assume that the end points of the interval are fixed under the fiber maps. Our paper thus extends work by V. Kleptsyn and D. Volk who treated step skew products where the fiber maps map the interval strictly inside itself. We clarify the dynamics for an open and dense subset of such skew products. In particular we prove existence of a finite collection of disjoint attracting invariant graphs. These graphs are contained in disjoint areas in the phase space called trapping strips. Trapping strips are either disjoint from the end points of the interval (internal trapping strips) or they are bounded by an end point (border trapping strips). The attracting graphs in these different trapping strips have different properties.

Stability as a Whole of a Family of Fibers Maps and Ω-Stability of C1-Smooth Skew Products of Maps of an Interval

Properties of C1 smooth skew products of maps of an interval with stable as a whole family of fibers maps are established. These results are applied to the proof of the criterion of Ω-stability of C1-smooth skew products of maps of an interval (with respect to homeomorphisms of skew products class).The proper subspace of the space of C1smooth skew products of maps of an interval is distinguished, where Ω-stable C1smooth skew products are contained. It is proved that Ω- stable skew products are not dense in the distinguished subspace of C1 -smooth maps.

Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity

We introduce a two-parameter family of ‘partially hyperbolic’ skew products (Ga, t)a > 0, t ∈ [ − ε, ε] maps with one dimensional centre direction. In this family, the parameter a models the central dynamics and the parameter t the unfolding of cycles (that occurs for t = 0). The parameter a also measures the ‘central distortion’ of the systems: for small a, the distortion of the systems is small and it increases and goes to infinity as a → ∞. The family (Ga, t) displays some of the main characteristic properties of the unfolding of heterodimensional cycles as intermingled homoclinic classes of different indices and secondary bifurcations via collision of hyperbolic homoclinic classes. For a ∈ (0, log 2), the dynamics of (Ga, t) is always non-hyperbolic after the unfolding of the cycle. However, for a > log 4 intervals of t-parameters corresponding to hyperbolic dynamics appear and turn into totally prevalent as a → ∞ (the density of ‘hyperbolic parameters’ goes to 1 as a → ∞). The dynamics of the maps Ga, t is described using a family of iterated function systems modelling the dynamics in the one-dimensional central direction.

Shadowing in Linear Skew Products

We consider a linear skew product with the full shift in the base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of shadowing for a typical pseudotrajectory of finite length. This result indicates that the high-dimensional analog of the Hammel–Yorke–Grebogi conjecture concerning the interval of shadowability for a typical pseudotrajectory is not correct. The main technique is the reduction of the shadowing problem to the ruin problem for a simple random walk.

Boundary Conditions for Maps in Fibers and Topological Transitivity of Skew Products of Interval Maps

We describe a general class of skew products of interval maps and obtain sufficient conditions for a skew product in this class to be topologically transitive, but not completely topologically transitive. Bibliography: 14 titles.