Abstract
In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation is obtained depending on parameters, which involves the Markov map topological entropy. The results are illustrated with examples.
Highlights
An important issue in the theory of forced systems is to evaluate the regularity of their synchronization graph
The orbits {(xt, yt)} of the skewproduct system (f, g) are attracted by the graph of the corresponding sync function.[2,17,19,22]. This sync function, say, φ : U → Y, can be defined by the conjugacy equation g[x, φ(x)] = φ ◦ f (x), ∀ x ∈ U, which ensures invariance of the corresponding graph {[x, φ(x)]}x∈U. Regularity properties of this function matter because they determine those dynamical characteristics of the drive system that carry over to the factor
The study of sync functions can be regarded as part of the analysis of inertial manifolds in dynamical systems.[12,13]
Summary
An important issue in the theory of forced systems is to evaluate the regularity of their synchronization graph. The study of sync functions can be regarded as part of the analysis of inertial manifolds in dynamical systems.[12,13] Besides existence and continuity, a standard result in this theory is the proof of differentiability under the condition The main result of this paper states that, for T a piecewise affine expanding and transitive Markov map, the total variation of φγ shows a sharp transition at γ = e−htop(T).
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