Symposium on Ergodic theory

- In honor of Professor Kyewon Koh Park's 65th birthday -

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"강연은 아래와 같은 순서로 진행됩니다."


Speaker: 김동한 (동국대)
Title: Diophantine approximation on translation surfaces
Abstract: Diophantine approximation deals with the approximation of real numbers by rational numbers. In geometric point of view, it corresponds to approximate a line on the plane of irrational slope by integral lattice points. We could understand Diophantine approximation using dynamics on the phase space by the recurrence time and also using dynamics on the moduli space or the parameter space by the escaping time to the cusp. In this talk, we consider Diophantine approximation on translation surfaces in two dynamical point of view and investigate the relation between them.


Speaker: 정의진 (아주대)
Title:
Expansive topological dynamical systems with uncountably many topologically transitive components
Abstract: A topological dynamical system (X,T) is called topologically transitive if it has a dense forward orbit. In many cases T is not topologically transitive, hence an invariant closed subset Y of X, together with the restriction of T, is called a (topologically transitive) component if (Y,T) is a maximal topologically transitive subsystem of (X,T). As the number of the subspaces of a topological dynamical system is at most cardinality of continuum, it is natural to ask whether there is a topological dynamical system with uncountably many topologically transitive components. After developing some machinery, in this paper we present a class of expansive Cantor homeomorphisms (shift spaces) with uncountably many topologically transitive components. The main ingredient is the closure of union of subsystems of a given system.


Speaker: 이정엽 (관동대)
Title:
Hexagonal aperiodic mono-tile tiling
Abstract: Aperiodic mono-tile (so called “einstein”) is a single prototile which tiles the plane only non-periodic way. After Penrose found the famous aperiodic tiling around 1974 consisting of two prototiles which tile the plane only non-periodically, it has been questioned among mathematicians and physicists that if there exists any aperiodic mono-tile. Around 2010, Taylor and Socolar came up with a single prototile which tiles the plane only non-periodically allowing to use the reflection of the prototile. Penrose noticed that it is quite similar with the tiling that he had built from (1+ε+ε²)-tiling. It turns out that the two tilings are not mutually locally derivable. But, in fact, there is some similarity and close relation. In this talk, we discuss how they are related with each other. Also we show why these tilings are always aperiodic.


Speaker: 손영환 (포스텍)
Title: Substitution dynamical systems
Abstract: A substitution is a map from a finite set of letters to the set of words, which can be used to produce an interesting infinite sequences. In this talk we will consider some statistical properties of dynamical systems arising from substitutions. Especially we would like to discuss mixing properties of substitution dynamical systems. If time permits, we also consider deviation problems of ergodic sums involving fixed points of substitutions.