arXiv subjects

Daniel J. Thompson

Publications and source records attributed to Daniel J. Thompson.

21 records · Page 2Linked to original sources

arXiv2012

Irregular sets, the $β$-transformation and the almost specification property

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $ϕ$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $ϕ$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $β$-shift satisfies almost specification and we show that the irregular set for any $β$-shift or $β$-transformation is either empty or has full topological entropy and Hausdorff dimension.

math.DS↗

arXiv2011

Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors

We give sufficient conditions for a shift space $(Σ,σ)$ to be intrinsically ergodic, along with sufficient conditions for every subshift factor of $Σ$ to be intrinsically ergodic. As an application, we show that every subshift factor of a $β$-shift is intrinsically ergodic, which answers an open question included in Mike Boyle's article "Open problems in symbolic dynamics". We obtain the same result for $S$-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.

math.DS↗

arXiv2009

A criterion for topological entropy to decrease under normalised Ricci flow

In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In this short note, we describe the problem and give a strong criterion under which monotonicity of the topological entropy can be established for a short time. In particular, the criterion applies to metrics of negative sectional curvature which are in the same conformal class as a metric of constant negative sectional curvature.

math.DS↗