After discussing usual approaches to measuring blur, we show theoretically that there is essentially a unique way to quantify blur by a single number and we confirm the usefulness of that measure by some experiment on a natural image.
We show that a class of robustly transitive diffeomor-phisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and… (More)
We introduce subshifts of quasi-finite type as a generalization of the well-known subshifts of finite type. This generalization is much less rigid and therefore contains the symbolic dynamics of many non-uniform systems , e.g., piecewise monotonic maps of the interval with positive entropy. Yet many properties remain: existence of finitely many ergodic… (More)
We show that a class of robustly transitive diffeomor-phisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have constant entropy and isomorphic unique measures of maximal entropy.
We study the dynamics of piecewise affine surface homeomor-phisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended… (More)
— We give a survey of the entropy theory of interval maps as it can be analyzed using ergodic theory, especially measures of maximum entropy and periodic points. The main tools are (i) a suitable version of Hofbauer's Markov diagram, (ii) the shadowing property and the implied entropy bound and weak rank one property, (iii) strongly positively recurrent… (More)
Extending work of Hochman, we study the almost-Borel structure, i.e., the non-atomic invariant probability measures, of symbolic systems and surface diffeomor-phisms. We first classify Markov shifts and characterize them as strictly universal with respect to a natural family of classes of Borel systems. We then study their continuous factors showing that a… (More)
It is known that piecewise affine surface homeomorphisms always have measures of maximal entropy. This is easily seen to fail in the discon-tinuous case. Here we describe a piecewise affine, globally continuous surface map with no measure of maximal entropy.