Invariant Graphs of Skew Product Dynamical Systems

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Abstract

We study real skew products Ëœ Tg(x,t) : X × R → X × R defined by Ëœ Tg(x,t) = (Tx,g(x,t)) where X is a compact Riemannian manifold, often the circle, T : X → X is an expanding Markov map or hyperbolic diffeomorphism and t 7→ g(x,t) is a homeomorphism. In particular we are interested in functions u : X →R satisfying Ëœ Tg(x,u(x)) = (Tx,u(Tx)), the graph of such a function is an invariant graph of the skew product. Initially, we study dynamical cohomology with respect to T : X → X a hyperbolic diffeomorphism or flow. That is, solutions u to the functional equation f = uT −u. Given a H¨older continuous function f : X → R, we prove that there exists an infinite sequence of Borel measurable functions un : X → R for n ∈ N such that f = u1T −u1 and un = un+1T −un+1 if and only if f = 0; building on the previous work of de Lima and Smania, and Bamo´n, Kiwi, Rivera-Letelier and Urzu´a. We then extend this to hyperbolic flows. We discuss an application of this problem to the regularity of invariant graphs of a family of skew products for which the invariant graphs are often the graphs of Weierstrass-type functions. Next, we consider skew products where T : X → X is an expanding Markov map of the circle such that Ëœ Tg : X ×R→ X ×R is non-uniformly expanding in thefibre ( R) coordinate. We study the case where there is a finite set of periodic orbits p ∈ P ⊂ X such that g(p,t) = t for all t ∈ R. That is, the skew product is the identity in the fibre coordinate for p ∈ P; and otherwise g(x,t) is expanding. We see that, under our conditions, there exists an essentially bounded invariant graph u and generically u is discontinuous on any open set. By joining the discontinuities with intervals, we define a unique invariant set that we call a quasi-graph. We calculate the box dimension of this set, showing that this extends the well known results of Bedford in the uniform case. In the final chapter, we allow the skew product to sometimes contract in the fibre direction but it must always expand on average. Under suitable conditions, the invariant graph is unbounded and a repeller of our skew product. We prove that the set of points in X×R repelled to−∞in the fibre is riddled with the set of points repelled to +∞ in a fractal manner. The stability index, defined by Ashwin and Podvigina, is a relatively unstudied function that measures the asymptotic rate at which two sets are locally riddled with one another. We calculate the stability index for points in the basins defined by these skew products and study the multi-fractal structure of the stability index

Keyword(s)

Coboundary; Dynamical cohomology; Ergodic Theory; Hyperbolic dynamics; Invariant Graphs; Skew product; Stability index; Weierstrass function

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