arXiv subjects

James P. Kelly

Publications and source records attributed to James P. Kelly.

4 recordsLinked to original sources

arXiv2021

Topological dynamics of Markov multi-maps of the interval

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 specification. To each Markov multi-map, we associate a shift of finite type (SFT), and then our main results relate the properties of the SFT with those of the Markov multi-map. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

math.DS↗

arXiv2019

Entropy conjugacy for Markov multi-maps of the interval

We consider a class $\mathcal{F}$ of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of $[0,1]^{\mathbb{Z}_+}$. For a multi-map in $\mathcal{F}$, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in $\mathcal{F}$.

math.DS↗

arXiv2019

Linear operators with infinite entropy

We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is shown that infinite topological entropy is equivalent to non-zero topological entropy for translation operators on weighted Lebesgue function spaces. In particular, finite non-zero entropy is impossible for this class of operators, which answers a question raised by Yin and Wei.

math.DS↗

arXiv2017

Direct Limits of Ad\`ele Rings and Their Completions

The ad\`ele ring $\mathbb A_K$ of a global field $K$ is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on $\mathbb A_K$. For a fixed global field $F$ and a possibly infinite algebraic extension $E/F$, there is a natural partial ordering on $\{\mathbb A_K:F\subseteq K\subseteq E\}$. Therefore, we may form the direct limit \[ \mathbb A_E = \varinjlim \mathbb A_K \] which provides one possible generalization of ad\`ele rings to arbitrary algebraic extensions $E/F$. In the case where $E/F$ is Galois, we define an alternate generalization of the ad\`eles, denoted $\bar{\mathbb V}_E$, to be a certain metrizable topological ring of continuous functions on the set of places of $E$. We show that $\bar{\mathbb V}_E$ is isomorphic to the completion of $\mathbb A_E$ with respect to any invariant metric and use this isomorphism to establish several topological properties of $\mathbb A_E$.

math.NT↗