# Dynamics seminar

Title: Cocycles and Skew Products

Speaker: Joseph Horan, University of Victoria

Date and time:
26 Nov 2015,
1:00pm -
1:50pm

Location: CLE C214

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NOTE THE EARLY START TIME PLEASE

Abstract: The idea of a cocycle is a natural one; we provide some motivating situations before proceeding to the definition of a cocycle. After some theoretical comments, we will explore how cocycles aid us in studying dynamical systems, in particular by forming natural skew products. The goal of the next talk is to describe the Multiplicative Ergodic Theorem, so we intend on covering the necessary background in this talk.

Title: A Lasota-York Inequality for GBTs

Speaker: Seth Chart, University of Victoria

Date and time:
19 Nov 2015,
1:30pm -
2:20pm

Location: CLE C214

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This week we will conclude with a definition of anisotropic spaces of densities adapted to GBTs and some key steps in the proof of a Lasota-Yorke inequality.

Title: Statistical properties of generalized baker's transformations via anisotropic Banach spaces.

Speaker: Seth Chart, University of Victoria

Date and time:
12 Nov 2015,
1:30pm -
2:20pm

Location: CLE C214

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Abstract: This week we will see what happens when we try to apply the methods developed for expanding interval maps to study generalized baker's transformations. These are piecewise hyperbolic maps of the unit square. We will see that the presence of both expanding and contracting coordinates precludes the use of standard spaces of functions to study the Frobenius-Perron operator. We will introduce anisotropic Banach space norms and see that we can recover the usual Lasota-Yorke inequality for the Frobinius-Perron operator acting on these spaces of anisotropic functions.

Title: Renewal Theory for an interval map

Speaker: Seth Chart, University of Victoria

Date and time:
05 Nov 2015,
1:30pm -
2:30pm

Location: CLE C214

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Abstract: This week we will look at what can be done when an interval map has an indifferent fixed point (T(x) =x and DT(x)=1). We will investigate some simple examples of generating function techniques and their application to renewal processes in probability theory. We will then discuss results of Sarig and Gouezel that extend this method to analyse Frobenius-Perron operators. We will try to see how this method works for a simple two branched map with onto branches and an indifferent fixed point at 0.

Title: Quasi-compactness for interval maps whose branches are not onto.

Speaker: Seth Chart, University of Victoria

Date and time:
29 Oct 2015,
1:30pm -
2:30pm

Location: CLE C214

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Last time Chris showed us that for piecewise Lasota-York maps the associated Frobenius-Perron (FP) operator can be viewed as an operator on C^1 functions and that this restriction is quasi-compact. Today we will investigate the effect of allowing the map T to have branches that are not onto. This small change in the map requires a substantial change in the analysis. We will see that the FP operator no longer restricts to C^1. Instead we will consider the restriction to functions of bounded variation (BV) and deduce quasi-compactness of the FP operator viewed as an operator on BV.

Title: Dynamical Limit Theorems via Spectral Methods

Speaker: Chris Bose, University of Victoria

Date and time:
22 Oct 2015,
1:30pm -
2:20pm

Location: CLE C214

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The classical CLT can be generalized to sequences of RV generated by a single observable and iterates of a measure preserving transformation. In particular, no independence is assumed, although some control on decay of correlation (for a class of functions containing the observable) is necessary. The method of proof involves the use of functional analytic versions of the classical characteristic functions from probability, something which seems to be interesting in its own right.

We are following a paper of Sebastien Gouezel which can be downloaded as item 5 from ‘other publications’ on the website

Title: Dynamical Limit Theorems, part III of III

Speaker: Chris Bose, University of Victoria

Date and time:
15 Oct 2015,
1:30pm -
2:20pm

Location: CLE C214

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The classical CLT can be generalized to sequences of RV generated by a single observable and iterates of a measure preserving transformation. In particular, no independence is assumed, although some control on decay of correlation (for a class of functions containing the observable) is necessary. The method of proof involves the use of functional analytic versions of the classical characteristic functions from probability, something which seems to be interesting in its own right.

We are following a paper of Sebastien Gouezel which can be downloaded as item 5 from ‘other publications’ on the website

https://perso.univ-rennes1.fr/sebastien.gouezel/publis.html

Title: The Dynamical Central Limit Theorem, Part II of III

Speaker: Chris Bose, University of Victoria

Date and time:
08 Oct 2015,
1:30pm -
2:30pm

Location: CLE C214

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The classical CLT can be generalized to sequences of RV generated by a single observable and iterates of a measure preserving transformation. In particular, no independence is assumed, although some control on decay of correlation (for a class of functions containing the observable) is necessary. The method of proof involves the use of functional analytic versions of the classical characteristic functions from probability, something which seems to be interesting in its own right.

It should be fairly easy to start with Part II since Part I was aimed at getting a feel for the classical result and its proof.

We are following a paper of Sebastien Gouezel which can be downloaded as item 5 from ‘other publications’ on the website