


Ergodic Theory
Org: Christopher Bose (Victoria) and Andres del Junco (Toronto) [PDF]
 WAEL BAHSOUN, University of Victoria
Deterministic representation for position dependent random
maps
[PDF] 
We give a deterministic skewtype representation for position
dependent random maps and describe the structure of the set of its
invariant densities. We prove onetoone correspondence between
absolutely continuous invariant measures (acims) for the position
dependent random map and the acims for its skew product
representation.
 ADAM FIELDSTEEL, Wesleyan University, Middletown, CT, USA
Relative pressure and the variational principle
[PDF] 
We define notions of pressure associated with topological and
measuretheoretic factors of compact dynamical systems and prove a
variational principle: The topological quantity is the supremum of the
set of the corresponding measuretheoretic quantities. A central
device is the use of average (or [`(d)]) separation of orbits in
the definition of pressure, in place of the classical notion of
separation. This change permits arguments that are, we hope,
simplifications of the classical arguments in the proofs of
variational principles, and which can readily be extended to obtain
analogous results for actions of amenable semigroups.
This is joint work with Russell Coe, CCSU.
 NIKOS FRANTZIKINAKIS, Penn State University
Sets of multiple recurrence
[PDF] 
An integer subset set S is a set of krecurrence if for every
measure preserving system and measurable set A with positive
measure, the sets A,T^{n}A,...,T^{kn}A intersect on a set of
positive measure for infinitely many n Î S. Furstenberg
constructed an example of a set of 1recurrence but not
2recurrence. For every positive integer k we will give explicit
examples of sets of krecurrence but not (k+1)recurrence (joint
work with E. Lesigne and M. Wierdl). We will also discuss the
question of whether the set of shifted primes is a set of
krecurrence (joint work with B. Host and B. Kra).
 EMILY GAMBER, UNCChapel Hill, CB #3250, Chapel Hill, NC 275993250
Topological Properties of Ddimensional Cellular Automata
[PDF] 
A cellular automaton is a tool used to model complex systems, making
discrete simulations of an intricate process. Cellular automata were
first investigated from a purely mathematical point of view in 1969
with Hedlund's formative paper. This work was motivated by
thencurrent problems in symbolic dynamics, possibly those of a
cryptographic nature. When Wolfram turned his attention to cellular
automata via computer simulation in the early 1980s, the subject
gained momentum. Wolfram categorized onedimensional cellular
automata based on features of their asymptotic behavior which could be
seen on a computer screen. Gilman's work in 1987 and 1988 was the
first attempt to mathematically formalize these characterizations of
Wolfram's; he utilized the notions of equicontinuity and
expansiveness, as well as measure theoretic analogs of each. While
measure is intrinsic to Gilman's partition,
Kurka
has a purely
topological classification centered on equicontinuity, expansiveness,
and sensitivity. We extend the onedimensional topological
classification of
Kurka
for cellular automata on the full shift
space, to higher dimensional subshift spaces, providing examples to
highlight the differences between one and twodimensional cellular
automata, as some results do not extend directly from one dimension.
 THIERRY GIORDANO, University of Ottawa, Ottawa
Connes and Woods Approximate Transitivity and Dimension
Spaces
[PDF] 
In 1985, Connes and Woods defined the notion of an approximately
transitive (AT) action of a group on a Lebesgue space in their study
of the flow of weights of particular von Neumann factors, the
ArakiWoods factors or equivalently of the associated flow of
ergodic, nonsingular transformations. Four years later, they studied
the Poisson boundary of groupinvariant, timedependent Markov random
walks defined on a locally compact group G and pointed out that it
is an AT and amenable Gspace. G. A. Elliott and myself proved that
any AT and amenable Gspace is such a Poisson boundary. With
D. Handelman, I have introduced the notion of dimension Gspaces to
study AT and amenable Gactions. In this talk, I will review the
notion of approximate transitivity and of dimension Gspaces and
present some specific constructions of AT and nonAT transformations.
 PAWEL GORA, Concordia University, Montreal
Properties of invariant densities of random maps
[PDF] 
We show a few results about the existence of absolutely continuous
invariant measure (acim) for random maps both with constant
probabilities and with probabilities dependent on position. Then, we
discuss some properties of the acim's. In particular we show an
example of a random map satisfying Pelikan's condition with the
support of acim consisting of infinite number of disjoint intervals
and with the density of acim not separated from 0. This is in
contrast with the properties of acim's for individual piecewise
expanding maps.
 GUANGYUE HAN, University of British Columbia
Analyticity of Hidden Markov Chains
[PDF] 
We prove that under a mild positivity assumption the entropy rate of a
hidden Markov chain varies analytically as a function of the
underlying Markov chain parameters. We give examples to show how this
can fail in some cases. And we study two natural special classes of
hidden Markov chains in more detail: binary hidden Markov chains with
an unambiguous symbol and binary Markov chains corrupted by binary
symmetric noise. Finally, we show that under the positivity
assumption the hidden Markov chain itself varies analytically,
in a strong sense, as a function of the underlying Markov chain
parameters.
 SHAFIQUL ISLAM, University of Lethbridge
Approximation of absolutely continious invariant measures for
random dynamical systems
[PDF] 
A random map is a discretetime dynamical system in which one of a
number of transformations is randomly selected and applied at each
iteration of the process. The asymptotic properties of a random map
are described by its invariant densities. I will talk about Ulam's
method for approximation of absolutely continious invariant measures
for Markov switching position dependent random maps. Piecewise linear
approximation of absolutely continious invariant measures for random
maps will also be discussed.
 DAVE MCCLENDON, University of Maryland
Orbit discontinuities of Borel semiflows on Polish spaces
[PDF] 
Let X be a standard Polish space. Given an action T_{t} of
[0,¥) by (presumably noninvertible) Borel maps on X, we say
that two distinct points x and y are "instantaneously
discontinuously identified" (IDI) if T_{t}(x) = T_{t}(y) for all t > 0. Such phenomena is an obstacle to representing the action as a
shift map on a space of continuous paths. We define the concept of
"orbit discontinuity", a generalization of IDI, and discuss results
regarding the structure and prevalence of such behavior. In
particular, the set of points which are IDI has measure zero with
respect to any measure preserved by the semiflow and is invariant
under Borel time changes.
This material is part of my Ph.D. research conducted under the
direction of Dan Rudolph.
 RANDALL MCCUTCHEON, University of Memphis, Memphis, TN 38152, USA
Central sets and multiple recurrence for nonamenable group
actions
[PDF] 
Measurable multiple recurrence results for nonnilpotent groups have
up to now been limited to an ergodic Roth theorem of Bergelson,
McCutcheon and Zhang, which states that for any measure preserving
actions {T_{g}}_{g Î G} and {S_{g}}_{g Î G} of a countable
amenable group G on a probability space (X,B,m) that
commute in the sense T_{g} S_{h} = S_{h} T_{g} for all g,h Î G, and any
A Î B with m(A) > 0, lim_{n} [ 1/(F_{n})]å_{g Î fn} m(AÇT_{g}^{1} AÇ(T_{g} S_{g})^{1} A) > 0
for any Følner sequence (F_{n}) for G. This yields, in
particular, that {g : m(AÇT_{g}^{1} AÇ(T_{g} S_{g})^{1} A) > 0} is syndetic. I'll be talking about some new techniques for
doing what might be called "ergodic theory without averaging" that
can be utilized to remove the amenability condition in this result
while simultaneously strengthening the conclusion.
 MARCUS PIVATO, Trent University, 2151 East Bank Drive, Peterborough,
Ontario K9L 1Z8
Crystallographic Defects in Cellular Automata
[PDF] 
Let A^{ZD} be the Cantor space of Z^{D}indexed
configurations in a finite alphabet A, and let s be the
Z^{D}action of shifts on A^{ZD}. A
cellular automaton is a continuous, scommuting selfmap
F of A^{ZD}, and a Finvariant subshift
is a closed, (F,s)invariant subset X Ì A^{ZD}. Suppose x Î A^{ZD} is Xadmissible
everywhere except for a small region we call a defect. It is
empirically known that such defects persist under iteration of F,
and propagate like `particles' which coalesce or annihilate when they
collide. We construct algebraic invariants for these defects, which
explain their persistence under F, and partly explain the
outcomes of their collisions. Some invariants are based on the
spectrum or cocyclestructure of X; others arise from the
higherdimensional (co)homology/homotopy groups of X, generalizing
methods of Conway, Lagarias, Geller, and Propp. We also study the
motion of defect particles (in the case D=1), and show that it falls
into several regimes, ranging from simple deterministic motion, to
random walks, to the emulation of Turing machines or pushdown
automata.
 IAN PUTNAM, University of Victoria
Orbit equivalence for Cantor minimal Z^{2}
systems
[PDF] 
The main result is that every free, minimal action of the group
Z^{2} on a Cantor set is orbit equivalent to an action of
Z. A complete invariant is given, which, for uniquely
ergodic actions, amounts to the values of the measure on the clopen
subsets.
This is joint work with Thierry Giordano, Hiroki Matui and Christian
Skau.
 ANTHONY QUAS, University of Victoria, Victoria, BC
SRB measures for expanding mappings
[PDF] 
We consider C^{1} expanding mappings of the circle and show that for a
residual set of such mappings (with respect to the C^{1} topology),
there is a unique SRB measure with basin of Lebesgue measure 1.
However, this measure is generically singular with respect to Lebesgue
measure.
This is joint work with James Campbell of the University of Memphis.
 JOE ROSENBLATT, Department of Mathematics, University of Illinois at
UrbanaChampaign, 273 Altgeld Hall, 1409 Green St., Urbana,
IL 61801, USA
Convergence and Divergence of Convolution Operators
[PDF] 
We consider R^{d} actions as groups of invertible measure
preserving transformations on probability spaces. There are many
interesting theorems and open questions related to the convergence and
divergence of sequences of operators determined by these actions, in
particular for those given by convolutions by L_{1}(R^{d})
functions. The behavior of these operators is closely tied to the
harmonic analysis of convolutions by these functions on L_{p}(R^{d}), 1 £ p £ ¥. Oscillation, rate of divergence, and
saturation theorems will be the focus of this talk.
 DAN RUDOLPH, Colorado State University, Fort Collins, Colorado
Kakutani equivalence in the finitary and Cantor minimal
categories: the beginnings of a theory
[PDF] 
The theory of Kakutani equivalence for measure preserving actions is
rich and well studied. In joint work with Nic Ormes, Wojciech Kosek
and Mrinal Roychowdhury we have begun to extend many of these ideas to
more topological realms. I will offer various definitions and the
collection of results we have been able to obtain so far.
 AYSE SAHIN, De Paul University, 2320 N. Kenmore Ave., Chicago, IL 60614,
USA
Directional Entropy, Rank One and Even Kakutani Equivalence
of Z^{d} Actions
[PDF] 
We present some recent results on the directional entropy of
Z^{d} rank one actions and their connection to Even Kakutani
Equivalence, an important example of a restricted orbit equivalence.
This is joint work with E. Arthur Robinson, Jr.
 CESAR SILVA, Williams College
Measurepreserving locally scaling transformations of
compactopen subsets of nonarchimedean local fields
[PDF] 
We introduce the notion of a locally scaling transformation defined on
a compactopen subset of a nonarchimedean local field. We show that
this class encompasses the Haar measurepreserving transformations
defined by C^{1} (in particular, polynomial) maps, and prove a
structure theorem for locally scaling transformations. We use the
theory of polynomial approximation on compactopen subsets of
nonarchimedean local fields to demonstrate the existence of ergodic
Markov, and mixing Markov transformations defined by such polynomial
maps. We also give simple sufficient conditions on the Mahler
expansion of a continuous map on the ring of padic integers Z_{p}
to Z_{p} for it to define a Bernoulli transformation.
This is joint work with J. Kingsbery, A. Levin, and A. Preygel.
 PIERRE TISSEUR, Trent University, Peterborough, Ontario
Inequalities, equalities between spatial and temporal
entropies of cellular automata
[PDF] 
A onedimensional cellular automaton F is a dynamical system on a
shift space X that can be defined by a local rule of radius r.
For a F and shift invariant measure m, the temporal entropy
h_{m} (F) depends on the way the automaton "moves" the spatial
entropy h_{m} (s). Using the discrete average Lyapunov
exponents I^{+}_{m} and I_{m}^{} we obtain for a shift ergodic and
Finvariant measure the inequality:
h_{m} (F) £ h_{m} (s) ×(I_{m}^{+} + I_{m}^{}). 

The exponents I_{m}^{} and I_{m}^{+} represent the lefttoright and
righttoleft average speeds of the faster perturbations. Taking in
account the average speed of all the perturbations, we obtain two
other equalities:
h_{m} (F) = h_{m} (s) × 
ó õ

X


lim
n®¥

M_{n}(I)(x)× 
I_{n}(x)
n

dm(x) 

and h_{m}(F) = h_{m} (s) ×M(r) ×r. The function
M_{n}(I)(x) represents for each x the proportion of perturbations
which propagate of I_{n}^{+}(x)+I_{n}^{}(x) = I_{n}(x) coordinates in n
iterations. The study of different examples shows that these
equalities and inequalities are good tools to show that the entropy of
a cellular automaton is equal to zero.
We have also some similar results for the Ddimensional cellular
automata (D > 1).
 REEM YASSAWI, Trent University, Peterborough, Ontario
Rank One measures which are not asymptotically randomised by
linear cellular automata
[PDF] 
Let X = {0,1}^{N} ® {0,1}^{N}. A
cellular automaton is a continuous map F on X which
commutes with the shift map s. In this talk F is the
linear automaton F(x) = x + s(x), with addition taken
componentwise modulo 2. We construct a family of rank one measures
m which satisfy the property that the weak star limit of
[ 1/(N)] å_{t=1}^{N} m°F^{t} is not the uniform
Bernoulli measure.

