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We prove that a periodic orbit $P$ with coprime over-rotation pair is an over-twist periodic orbit iff the $P$-linear map has the over-rotation interval with left endpoint equal to the over-rotation number of $P$. We then show that this result fails if the over-rotation pair of $P$ is not coprime. Examples of patterns with non-coprime over-rotation pairs are given so that these patterns have no block structure over over-twists but have over-rotation number equal to the left endpoint of the forced over-rotation interval (such patterns are called emph{very badly ordered}). This presents a situation in which the results about over-rotation numbers on the interval and those about classical rotation numbers for circle degree one maps are different. In the end we elucidate a rigorous description of the strongest unimodal pattern that corresponds to a given over-rotation interval and use it to construct unimodal very badly ordered patterns with arbitrary non-coprime over-rotation pair.
We describe all possible bimodal over-twist patterns. In particular, we give an algorithm allowing one to determine what the left endpoint of the over-rotation interval of a given bimodal map is. We then define a new class of polymodal interval maps
For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniforml
We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we investigate whether they have various properties, including topological transitivity, topological mixing, dense periodic points, and specifica
In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for
For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^ntoR$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation pro