Exact bounds for thepolynomial decay of correlation, 1/f noise and theCLT for the equilibrium state of a non-Hölder potential

Abstract

We analyse the correlation and limitbehaviour of partial sums for thestationary stochastic process (f(Tt(x)),µ),t = 0,1,..., forfunctions f of superpolynomialvariation, the class \U0001d4ae\U0001d4abdefined below (which includes the Hölderfunctions), where T:Σ+→Σ+ is the left shift map on Σ+ = Π0∞{0,1} and µ isthe non-atomic equilibrium measure of a non-Hölder potential g = gγ belonging to a one-parameterfamily, indexed by γ>2.First, using the renewal equation, we show a polynomial rate ofconvergence for the associated Ruelle operator for cylinder setobservables.We then use these estimates to prove the following theorems:• We extend the polynomial convergence for the Ruelleoperator to functions f∊\U0001d4ae\U0001d4ab.• We show that the measure is weaklyBernouilli and the bounds are polynomial.• We calculate the decay of correlation of the stationarystochastic process described above, for f∊\U0001d4ae\U0001d4ab. This decay is polynomialwith t: we show in theorem 4.1 an upperbound of the order of Ct2-γ whenγ>2;this estimate is sharp in the sense that for each γ there existfunctions f(in fact f = I[0] gives an example)for which one has the lower bound of ct2-γ for the decay ofits autocorrelation(see theorem \\ref{t:decay and 1/fnoise}). For the lower bound we useTauberian theorems. For this example the coefficients decaymonotonically, which is important for proving the lower bound.• Again using Tauberian theorems together with theupper-lower bounds we show that for each 3>γ>2 one has thephenomenon of 1/f noise for the spectral density of the function I[0](see theorem \\ref{t:decay and 1/fnoise}).• We prove the central limit theorem (CLT)and functional CLT for the case where fis in\U0001d4ae\U0001d4ab and forγ>3 (theorem \\ref{t:CLT}). For thiswe apply Gordin's method in the setting ofa polynomial rate of convergence.From the perspective of differentiable dynamical systems µ is theunique invariant measure which isabsolutely continuous with respect to theLebesgue measure for an associateddoubling map of the circle with anindifferent fixed point. This map T1 = T1,γ is a piecewise linearversion of the Manneville-Pomeau map, andthe potentialgγ is equal to-log DT1,γ.We emphasize that our class \U0001d4ae\U0001d4ab is larger than the classes studiedelsewhere.