arXiv subjects

Jonathan Meddaugh

Publications and source records attributed to Jonathan Meddaugh.

22 records · Page 2Linked to original sources

arXiv2020

Expansivity and unique shadowing

Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing and two-sided s-limit shadowing are equivalent. We show that $f$ is positively expansive and has shadowing if and only if it has unique shadowing (i.e.\ each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter's proof that positively expansive maps with shadowing are topologically stable. We use the aforementioned result on two-sided shadowing to find an equivalent characterisation of shadowing and expansivity and extend these results to the notion of $n$-expansivity due to Morales.

math.DS↗

arXiv2019

Limit Sets and Internal Transitivity in Free Group Actions

It has been recently shown that, under appropriate hypotheses, the $ω$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the action of a finitely generated free group or monoid. We give general definitions for several types of limit sets and analogous notions of internal transitivity. We then demonstrate that these limit sets are completely characterized by internal transitivity properties in shifts of finite type and general dynamical systems exhibiting a form of the shadowing property.

math.DS↗

arXiv2017

Shifts of finite type as fundamental objects in the theory of shadowing

Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let $X$ be a compact totally disconnected space and $f:X\to X$ a continuous map. We demonstrate that $f$ has shadowing if and only if the system $({f},{X})$ is (conjugate to) the inverse limit of a directed system of shifts of finite type. In particular, this implies that, in the case that $X$ is the Cantor set, $f$ has shadowing if and only if $(f,X)$ is the inverse limit of a sequence of shifts of finite type. Moreover, in the general compact metric case, where $X$ is not necessarily totally disconnected, we prove that $f$ has shadowing if and only if $({f},{X})$ is a factor of (i.e. semi-conjugate to) the inverse limit of a sequence of shifts of finite type by a quotient that almost lifts pseudo-orbits.

math.DS↗

arXiv2008

Any counterexample to Makienko's conjecture is an indecomposable continuum

Makienko's conjecture, a proposed addition to Sullivan's dictionary, can be stated as follows: The Julia set of a rational function R has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko's conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational functions whose Julia set is an indecomposable continuum.

math.DS↗