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4:00 pm Wednesday, March 18, 2009 Geometry-Analysis Seminar: Fat Solenoidal Attractorsby
Masato Tsujii (University of Kyushu) in HB 227- We begin with considering a simple (but chaotic) dynamical system $F:S1\times R \to S1\times R$, $(x,y) \mapsto (\ell x, \lambda y+f(x))$, where $\ell \ge 2$ is an integer, $0<\lambda<1$ a real number, $f(x)$ a smooth function. This is a skew product of contractions on the line over an expanding map on the circle, so that $F$ admits an attractor $A$ and a physical measure $\mu$. If $\lambda$ is small, the attractor $A$ locally looks like the product of a thin Cantor set and a curve. As $\lambda$ becomes large and close to $1$, the attractor $A$ becomes "fat" and $\mu$ becomes smooth, under some generic conditions on $f(x)$. We give a few statements that describe these changes precisely. Then we discuss how such results can be extended to more general situations including partially hyperbolic dynamical systems. Finally we would like to give a few conjectures concerning the singular supports of the SBR measures for partially hyperbolic dynamical systems and discuss about its relation to the so-called stable ergodicity conjecture.
Submitted by damanik@rice.edu |