The Final Exam will be on Saturday 13 March 2010 at 11:30am-2:30pm in WG YOUNG CS24.

Mixing on Rank-One Transformations
Darren Creutz and Cesar Silva
Studia Mathematica (to appear)

We prove mixing on rank-one transformations is equivalent to “the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums”. In particular, all staircase transformations (and polynomial staircase transformations) are mixing, answering Adams' question.

The Midterm will be on 15 February 2010 during class.

The Final Exam will be on 13 March 2010 at 11:30am-2:20pm in the usual rooms.

Teaching assistant for Math 115A Linear Algebra (Winter 2010) at UCLA with Professor Yehuda Shalom.

Superstability and Finite Time Extinction for C₀-Semigroups
D. Creutz, M. Mazo Jr. and C. Preda
Submitted

A new approach to superstability and finite time extinction of strongly continuous semigroups is presented, unifying known results and providing new criteria for these conditions to hold analogous to the well-known Pazy condition for stability. That finite time extinction implies superstability which is in turn equivalent to several (both known and new) conditions follow from this new approach in a consistent fashion. Examples that the converse statements fail are constructed, in particular, an answer to a question of Balakrishnan on superstable systems not exhibiting finite time extinction.

Teaching assistant for Math 111 Theory of Numbers (Spring 2009) at UCLA with Professor Yehuda Shalom.

Mixing on Rank-One Transformations
Darren Creutz and Cesar Silva
Submitted

We prove mixing on rank-one transformations is equivalent to “the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums”. In particular, all staircase transformations (and polynomial staircase transformations) are mixing, answering Adams' question.

Mixing on a Class of Rank-One Transformations
Darren Creutz and Cesar Silva
Journal of Ergodic Theory and Dynamical Systems

We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences. The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations implying in particular Adams’ result on staircase transformations. Another application yields a new proof that Ornstein’s class of rank-one transformations constructed using “random spacers” are almost surely mixing transformations.

Rank-One Mixing and Dynamical Averaging
Darren Creutz
Honors Thesis (Williams College)

A rank-one transformation is defined by a sequence of positive integers, the sequence of cuts, and a dynamical sequence of nonnegative integers, the sequence of spacers, that are used to repeatedly cut and stack a single column. Our main result is that rank-one transformations satisfying a condition called restricted growth and such that the spacer sequence is uniformly ergodic with respect to the transformation are mixing transformations. This result and the related results presented in this exposition are to be presented in the paper above. The restricted growth condition limits the total variation in the spacer sequence and is a generalization of a condition, equivalent to restricted growth for staircase transformations, given by Adams, that is sufficient for staircase transformations to be mixing, while the uniform ergodicity of the spacer sequence is a generalization of the notion of uniform Cesaro transformations used by Adams to show mixing on staircases. The application of our concepts and results to a class of rank-one transformations, a class we call generalized staircase transformations, yields a variety of rank-one mixing transformations with explicit constructions.