Research Interests:
My research interest lies primarily in applying topological methods to dynamical systems. Of particular interest is the notion of shadowing — i.e. how well an orbit calculated with finite precision represents the actual behavior of a dynamical system.
I am also very keenly interested in the study of continua, especially those arising as inverse limits of dynamical systems.
Research Preprints:
- Shifts of finite type as fundamental objects in the theory of shadowing.
Inventiones Mathematicae (2019). Joint with C. Good - Shadowing is generic on dendrites. Discrete and Continuous
Dynamical Systems Series S, 12 (2019), no.8, 2211-2220. Joint with W.R. Brian and B.E. Raines - Weak specification and Baire space. Journal of Mathematical Analysis
and Applications 479 (2019), no.1, 1355-1363. Joint with B.E. Raines - Orbital shadowing, internal chain transitivity and ω-limit sets. Ergodic
Theory and Dynamical Systems 38 (2018), no.1, 143-154. Joint with C. Good - Invariant measures on multi-valued functions. Journal of
Mathematical Analysis and Applications 456 (2017), no. 1, 616-627. Joint with B.E. Raines and T. Tennant - Chain transitivity and variations of the shadowing property. Ergodic Theory and Dynamical Systems 35 (2015), no. 7, 2044-2052. Joint with W.R. Brian and B.E. Raines
- Convergence of sequences of inverse limits. Topology and its Applications
184 (2015), 29-40. Joint with J.P. Kelly - Chaos for successive maxima map imples chaos for the original map. Nonlinear Dynamics 79 (2015), no. 3, 2165-2175. Joint with A. Boyarsky, P. Eslami, P. Gora, Z. Li, and B.E. Raines
- Shadowing and ω-limit sets of circular Julia sets. Ergodic
Theory and Dynamical Systems 35 (2015), no. 4, 1045-1055. Joint with A.D. Barwell and B.E. Raines - Shadowing and internal chain transitivity. Fundamenta Mathematicae 222 (2013), no. 3, 279-287. Joint with B.E. Raines
- Indecomposability in inverse limits with set-valued functions. and its Applications 160 (2013), no. 13, 1720-1731. Joint with J.P. Kelly
- The structure of limit sets for Zd actions. Discrete and Continuous
Dynamical Systems 37 (2014), no. 11, 4765-4780. Joint with B.E. Raines - Any counterexample to the Makienko conjecture is an indecomposable continuum. Ergodic Theory and Dynamical Systems 29 (2009),
no. 3, 875-883. Joint with C.P. Curry, J.C. Mayer, and J.T. Rogers, Jr. - Embedding one-dimensional continua into T×I. and its Applications 153
(2007), no. 18, 3519-3527.