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By employing the recurrence method worked out in Pawelec (2019 arXiv: 1911.12992), we provide effective lower estimates of the proper–dimensional Hausdorff measure of minimal sets of circle homeomorphisms that are not conjugate to any rotation.
Citations
... We discuss this on an easy example. An upcoming paper with M. Urbański [4] shows a more interesting application, namely for Cantor sets defined by the so-called Denjoy maps (i.e. we show a bound from below on the Hausdorff measure of the minimal set occurring for a C 1+α diffeomorphism on the circle which is only semi-conjugate to a rotation). ...
... If a system has slow recurrence properties, then one could get meaningful results as well. An example of this is in a paper with M. Urbański [4], where the underlying dynamics is that of an irrational rotation on a circle (which for numbers with bad Diophantine properties is in fact slowly recurrent). ...
We show a new method of estimating the Hausdorff measure of a set from below. The method requires computing the subsequent closest return times of a point to itself.
