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Dynamical Systems and Related Topics Workshop
March 21-24, 1998
University of Maryland
College Park, MD, USA

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Mike Boyle, Brian Hunt, Jim Yorke

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Statistical properties for multi-dimensional intermittent maps
by
Michiko Yuri
Sapporo University

Statistical properties for multi-dimensional intermittent maps.

Statistical properties for multi-dimensional intermittent maps.

Michiko Yuri
Department of Business Administration
Sapporo University
JAPAN

1  ABSTRACT

We shall discuss statistical properties of multi-dimensional noninvertible maps admitting indifferent periodic points ( which are so-called intermittent maps). More precisely, let T be a nonsingular transformation on a nonatomic probability space (X, F, \nu) (i.e., \nu ~ \nuT-1). We assume that there exists a countable generating partition Q = { Xa }a in I of X such that for every Xa in Q, \nu(Xa) > 0 and T|Xa: Xa --> T Xa is an invertible map satisfying [(d(\nuT)|Xa)/(d\nu|Xa)] > 0 (\nu - a.e). The generating partition Q does not necessarily satisfy the Markov property with respect to T but satisfies the FRS condition from which we can obtain nice countable state symbolic dynamics which provide natural Markovization of (T, X, Q) ([6]). T admits certain nonhyperbolic periodic orbits which may cause interesting phenomena from statistical point of view.
Definition
A periodic point x0 (with period q ) is called indifferent with respect to \nu if [(d(\nuTq))/(d\nu)](x0) = 1 (cf.[8].)
In particular, for piecewise C1 maps T and for \nu the Lebesgue measure, the definition coincides with the usual one (i.e., |detDTq(x0)| = 1). For such intermittent maps, the existence and further ergodic properties of T -invariant measure \mu <= \nu were obtained in previous papers [4-5]. In this talk, we shall give a class of functions in which the central limit theorem holds with respect to \mu ([7]). For this purpose first we clarify the speed of L1-convergence of iterated Perron-Frobenius operators by usying their random perturbations (cf.[2]). Then we extended the results on the central limit theorem for Markov fibred systems ([1]) to more general class of intermittent maps which contains our examples below. Furthermore, in case when T is a piecewise C1 -intermittent map we shall prove large deviation results for preimages weighted by the derivatives (i.e., upper bounds in the level 2 large deviation principle) under certain conditions ([3]).
Example(1).
( Diophantine approximation algorism in Inhomogeneous linear class). Let X = {(x1, x2) in R2 : 0 <= x2 <= 1, -x2 <= x1 <= - x2 + 1 }. T is defined on X by
T(x1, x2) = ( 1/x1 - [(1 - x2)/x1] + [-(x2/x1)], -[-(x2/x1)] - (x2/x1)).
Let I = {(a, b) in Z2 :a > b > 0, a < b < 0 } and for each (a, b) in I define
X(a, b) = { (x1, x2) in X : [(1-x2)/x1] - [ -(x2/x1)] = a, -[-(x2/x1)] = b }.
Then Q = { X(a, b) }(a, b) in I is a countable generating partition of X and on each X(a, b), T is given by T(x1, x2) = ( 1/x1 - a, -(x2/x1) + b ). The points (1, 0) and (-1, 1) are indifferent periodic points with period 2.
Example(2).
(Brun's algorism). Let X = { (x1, x2) in R2 :0 <= x2 <= x1 <= 1 } and let Q = { Xi }i=02, where Xi = {(x1, x2) in X : xi + x1 >= 1 >= xi+1 + x1 } and x0 = 1, x3 = 0. T is defined by T|X0(x1, x2) = ( [(x1)/(1-x1)], [(x2)/(1-x1)]), T|X1(x1, x2) = ( [ 1/(x1)] - 1, [(x2)/(x1)]), T|X2(x, y) = ( [(x2)/(x1)], [ 1/(x1)] - 1 ). T admits one indifferent fixed point (0, 0) .
References
1  J.Aaronson, M.Denker & M.Urbanski. Ergodic Theory for Markov fibred systems and parabolic rational maps. Trans.AMS, 337 (1993), 497-548.
2  C.Liverani. Flows, random perturbations and rates of mixing. To appear in Ergodic Theory and Dynamical Systems.
3  M.Pollicott, R.Sharp & M.Yuri. Large deviations for maps with indifferent fixed points. To appear in Nonlinearity.
4  M.Yuri. On a Bernoulli property for multi-dimensional maps with finite range structure. Tokyo J.Math. 9 (1986), 457-485.
5  M.Yuri. Invariant measures for certain multi-dimensional maps. Nonlinearity, 7 (1994), 1093-1124.
6  M.Yuri. Multi-dimensional maps with infinite invariant measures and countable state sofic shifts. Indagationes Mathematicae, 6, (1995) 355-383.
7  M.Yuri. Statistical properties for nonhyperbolic maps with finite range structure. Preprint.
8  M.Yuri. Thermodynamic Formalism for certain nonhyperbolic maps. Preprint.

Date received: March 10, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabf-19.