Physical Billiards and Finite Time Predictions for First Passage Probabilities

Series: 
Dissertation Defense
Thursday, March 15, 2018 - 13:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Georgia Tech
,  
I will discuss two topics in Dynamical Systems.  A uniformly hyperbolic dynamical system preserving Borel probability measure μ is called fair dice like or FDL if there exists a finite Markov partition ξ of its phase space M such that for any integers m and j(i), 1 ≤ j(i) ≤ q one has μ ( C(ξ, j(0)) ∩ T^(-1) C(ξ, j(1)) ∩ ... ∩ T^(-m+1)C(ξ, j(m-1)) ) = q^(-m) where q is the number of elements in the partition ξ and C(ξ, j) is element number j of ξ.  I discuss several results about such systems concerning finite time prediction regarding the first hitting probabilities of the members of ξ.  Then I will discuss a natural modification to all billiard models which is called the Physical Billiard.   For some classes of billiard, this modification completely changes their dynamics.  I will discuss a particular example derived from the Ehrenfests' Wind-Tree model.  The Physical Wind-Tree model displays interesting new dynamical behavior that is at least as rich as some of the most well studied examples that have come before.