Researcher Information


Associate Professor

The Mystery of the Edge of Chaos

Department of Mathematics, Mathematics


Ergodic theory of nonhyperbolic dynamical systems

FieldErgodic theory, dynamical systems theory
KeywordNonhyperbolic dynamical systems, transfer operator, metastability, homoclinic tangency, random dynamical systems

Introduction of Research

Dynamical systems theory is an area for the study of what happens after a long period of time given an evolution
law per unit time, such as a sequence with a recurrence relation or the solution of an autonomous differential
equation (the corresponding time action is called a dynamical system). It need not be limited to models in mechanics,
but covers mathematical models of various natural and social phenomena such as population, stock prices, and precipitation.

In many dynamical systems such as weather models, a small change in the initial state can have a large impact after
a long period of time (it may make sense to consider the fact that weather forecasts for the next year are not always accurate).
Such a phenomenon is called "chaos". Chaos makes it difficult to predict the future in the level trajectories, but from a statistical
point of view, chaos rather makes analysis easier. On the other hand, the statistical properties of dynamical systems on the edge
of chaos, such as turbulence, are often not well understood even today.

For dynamical systems with a good differential geometrical property, called hyperbolicity, it is known to be isomorphic to chaotic
symbolic dynamical systems, and therefore to have various good statistical properties such as the law of large numbers,
the central limit theorem, and the existence and uniqueness of equilibrium measures. My specialty is nonhyperbolic dynamical systems
(i.e. dynamical systems being not hyperbolic), which says almost nothing because there are several different types of nonhyperbolic
dynamical systems. I have studied, or am currently interested in, partially hyperbolic dynamical systems with neutral directions (e.g. Lorentz gas),
infinite ergodic chaos derived from neutral fixed points (e.g. intermittency in Pomeau-Manneville type turbulence models),
dynamical systems with homoclinic tangency (e.g. three-species competing model without time averages), and so on. I'm also interested
in the differences between variants of dynamical systems that have emerged in response to demands from other natural sciences
(e.g. random dynamical systems, coupled map lattice) and conventional dynamical systems.