We study real skew products Ë Tg(x,t) : X Ã R â X Ã R deï¬ned by Ë Tg(x,t) = (Tx,g(x,t)) where X is a compact Riemannian manifold, often the circle, T : X â X is an expanding Markov map or hyperbolic diï¬eomorphism and t 7â g(x,t) is a homeomorphism. In particular we are interested in functions u : X âR satisfying Ë 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 â X a hyperbolic diï¬eomorphism or ï¬ow. That is, solutions u to the functional equation f = uT âu. Given a HÂ¨older continuous function f : X â R, we prove that there exists an inï¬nite sequence of Borel measurable functions un : X â R for n â N such that f = u1T âu1 and un = un+1T âun+1 if and only if f = 0; building on the previous work of de Lima and Smania, and BamoÂ´n, Kiwi, RiveraLetelier and UrzuÂ´a. We then extend this to hyperbolic ï¬ows. 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 Weierstrasstype functions. Next, we consider skew products where T : X â X is an expanding Markov map of the circle such that Ë Tg : X ÃRâ X ÃR is nonuniformly expanding in theï¬bre ( R) coordinate. We study the case where there is a ï¬nite set of periodic orbits p â P â X such that g(p,t) = t for all t â R. That is, the skew product is the identity in the ï¬bre coordinate for p â 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 deï¬ne a unique invariant set that we call a quasigraph. We calculate the box dimension of this set, showing that this extends the well known results of Bedford in the uniform case. In the ï¬nal chapter, we allow the skew product to sometimes contract in the ï¬bre 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 XÃR repelled toââin the ï¬bre is riddled with the set of points repelled to +â in a fractal manner. The stability index, deï¬ned 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 deï¬ned by these skew products and study the multifractal structure of the stability index
Date of Award  1 Aug 2017 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Nikita Sidorov (Supervisor) & Charles Walkden (Supervisor) 

 Ergodic Theory
 Dynamical cohomology
 Coboundary
 Stability index
 Weierstrass function
 Skew product
 Invariant Graphs
 Hyperbolic dynamics
Invariant Graphs of Skew Product Dynamical Systems
Withers, T. (Author). 1 Aug 2017
Student thesis: Phd