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The separation time s(x, y) enables us to define a metric on that measures the distance between x and y. Definition 12. (Metric on the Tower) For x, y ∈ we define d(x, y) = βs(x,y), with β arbitrarily chosen in (0, 1), to be the metric that captures the symbolic distance of x and y on the tower . One can verify that this is indeed a metric. In particular, what makes this a metric is the fact that the partition is separating. No pair of points will stay together forever and, therefore, this gives βs(x,y) = 0 ⇔ x = y. 1.2.2 Regularity conditions on the Markov Tower We assume that F carries m l,i to m l+1,i without distortion if l < Ri − 1 and hence, if we denote the Jacobian of F by JF, JF = 1 on \ F−1 0. For the top of the tower, F−1 0, we impose a regularity condition on JFF−1 0 or equivalently on JFR 0 as follows: The return map restricted on each element of the partition, FR 0,i : 0,i → 0, and its local inverses are measurable and nonsingular with respect to the reference measure m so that measures with densities with respect to the reference measure m are transformed to measures with densities with respect to m under F. Additionally, as a regularity condition on the Jacobian on the top of the Tower, we ask that log JFR 0 be uniformly Lipschitz as follows. There exist C > 0 and 0 < β < 1 fixed such that JFR(x) JFR(y) − 1 ≤ Cd(x, y) = Cβs(x,y) for all i and for all x, y ∈ 0,i. (1.20) As we mentioned earlier JF(x) is a measure of expansion or contraction at the point x under FR so how far the ratio of the derivatives at x and y is away 14
Object Description
Title  Mixing conditions and return times on Markov Towers 
Author  Psiloyenis, Yiannis 
Author email  psiloyen@usc.edu 
Degree  Doctor of Philosophy 
Document type  Dissertation 
Degree program  Mathematics 
School  College of Letters, Arts and Sciences 
Date defended/completed  20080523 
Date submitted  2008 
Restricted until  Unrestricted 
Date published  20080624 
Advisor (committee chair)  Haydn, Nicolai 
Advisor (committee member) 
Baxendale, Peter H. Jonckheere, Edmond A. Lototsky, Sergey Vladimir 
Abstract  This dissertation discusses mixing properties derived on nonuniformly hyperbolic dynamical systems which admit a MarkovTower structure. For systems with exponential and polynomial decay of the tails of the return map we derive alphamixing conditions at exponential and polynomial rates, respectively. The motivation is to use these mixing conditions, on eligible systems, to approximate the law of the hitting and return times to a set of small measure.; In the first chapter we set up the problem and derive mixing conditions. In the next chapter we give a brief discussion on Stein method and in chapter three we study the hitting and multiplereturn times for αmixing dynamical systems in general. Under the given rates of mixing we use the Stein method to show that the return time of order k to a cylinder A can be approximated by a simple distribution for which sharp error bounds, independent of the order k, are obtained as well. Additionally, we conclude that the distribution of hitting times, suitably rescaled, can be approximated by an exponential distribution with mean 1.; As an application we show that the findings for the return and hitting times can be applied, through the construction of aMarkov Tower, to the GaspardWang map which is broadly used in Physics. 
Keyword  Stein method; Poisson approximation; hyperbolic dynamical systems; Markov towers; hitting time; return times; mixing 
Language  English 
Part of collection  University of Southern California dissertations and theses 
Publisher (of the original version)  University of Southern California 
Place of publication (of the original version)  Los Angeles, California 
Publisher (of the digital version)  University of Southern California. Libraries 
Type  texts 
Legacy record ID  uscthesesm1288 
Contributing entity  University of Southern California 
Rights  Psiloyenis, Yiannis 
Repository name  Libraries, University of Southern California 
Repository address  Los Angeles, California 
Repository email  cisadmin@lib.usc.edu 
Filename  etdPsiloyenis20080624 
Archival file  uscthesesreloadpub_Volume44/etdPsiloyenis20080624.pdf 
Description
Title  Page 20 
Contributing entity  University of Southern California 
Repository email  cisadmin@lib.usc.edu 
Full text  The separation time s(x, y) enables us to define a metric on that measures the distance between x and y. Definition 12. (Metric on the Tower) For x, y ∈ we define d(x, y) = βs(x,y), with β arbitrarily chosen in (0, 1), to be the metric that captures the symbolic distance of x and y on the tower . One can verify that this is indeed a metric. In particular, what makes this a metric is the fact that the partition is separating. No pair of points will stay together forever and, therefore, this gives βs(x,y) = 0 ⇔ x = y. 1.2.2 Regularity conditions on the Markov Tower We assume that F carries m l,i to m l+1,i without distortion if l < Ri − 1 and hence, if we denote the Jacobian of F by JF, JF = 1 on \ F−1 0. For the top of the tower, F−1 0, we impose a regularity condition on JFF−1 0 or equivalently on JFR 0 as follows: The return map restricted on each element of the partition, FR 0,i : 0,i → 0, and its local inverses are measurable and nonsingular with respect to the reference measure m so that measures with densities with respect to the reference measure m are transformed to measures with densities with respect to m under F. Additionally, as a regularity condition on the Jacobian on the top of the Tower, we ask that log JFR 0 be uniformly Lipschitz as follows. There exist C > 0 and 0 < β < 1 fixed such that JFR(x) JFR(y) − 1 ≤ Cd(x, y) = Cβs(x,y) for all i and for all x, y ∈ 0,i. (1.20) As we mentioned earlier JF(x) is a measure of expansion or contraction at the point x under FR so how far the ratio of the derivatives at x and y is away 14 