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The dichotomy for specific systems
One of the results in [FFT12] is that for uniformly expanding systems like the doubling map there is a dichotomy in terms of the type of laws of rare events that one gets at every possible centre . Namely, it was showed that either is non-periodic in which case one always gets a standard exponential EVL/HTS or is a periodic (repelling) point and one obtains an exponential law with an EI 0 < < 1 given by the expansion rate at (see [FFT12, Section 6]). This was proved for cylinders rather than balls, meaning that the set Un’s are dynamically defined cylinders (see [FFT12, Section 5] or [FFT11, Section 5], for details). Results for cylinders are weaker than the ones for balls, since, in rough terms, it means that the limit is only obtained for certain subsequences of n N rather than the whole sequence. In [FFT12], it was conjectured that this dichotomy should hold in greater generality, namely for balls rather than cylinders and for more general systems. As a consequence of Theorem A we will be able to show both. We remark that from the results in [FP12], one can also derive the dichotomy for conformal repellers and, in [K12], the dichotomy is also obtained for maps with a spectral gap for their Perron-Frobenius operator. In both these papers, the results were obtained by studying the spectral properties of the Perron-Frobenius operator. Now, we will give some examples of systems to which we can apply Theorem A in order to prove a dichotomy regarding the existence of an EI equal to 1 or less than 1. Moreover, we will see that Corollary C yields the second, and more general, aspect of this dichotomy: convergence type of REPP, namely, standard or compound Poisson. In its both aspects, the dichotomy depends on the centre being a non-periodic or periodic point, respectively. We will study the dichotomy for uniformly expanding and piecewise expanding maps, when all points in the orbit of are continuity points of the map.
The extremal behaviour at discontinuity points
In this section, we go back to Rychlik maps introduced in Section 3.3.1, however this time we consider only the ones with finitely many branches, and study the extremal behaviour of these systems when the orbit of hits a discontinuity point of the map. Consider a point Y . Note that here we have at most finitely many collection of open intervals such that i Y i Y . If Λ then we say that is a simple point . If is a non- simple point , which means that rS ( ) is finite, then let = rS ( ) and z = f ( ). We will always assume that z S is such that: there exist i+, i N so that z is the right end point of Yi and the left end point of Yi+ . We consider that the point z is doubled and has two versions: z+Yi+ and z Yi , so that f (z+) := fi+ (z) = limx z, xY i + f (x) and f (z ) := fi (z) = limx z, x Yi f (x). Whenis a non-simple point we consider that its orbit bifurcates when it hits S and consider its two possible evolutions. We express this fact by saying that when is non-simple we consider the “orbits” of + and
which are defined in the following way:
• for j = 1, . . . , we let f j ( ±) := f j ( );
• for j = + 1, we define f j ( ±) := fi± (f j1 ( ±))
• for j > + 1 we consider two possibilities:
– if j 1 is such that f j1 ( ±) / S, then we set f j ( ±) := f (f j 1( ±))
– otherwise we set f j ( ±) := fi(f j1 ( ±)), where i is such that f j (z±)Yi.
Laws of rare events for stochastic dynamics
In this section we will start with the proof of Theorem D which states that the dichotomy observed in Section 3.2 vanishes when we add absolutely continuous (with respect to Lebesgue) noise to the original system and for every chosen point in the phase space we have a standard exponential distribution for the EVL and HTS/RTS weak limits. We will also certify that the REPP converges to a Poisson Process with intensity 1. Next, we will give some examples of random dynamical systems for which we can prove the existence of EVLs and HTS/RTS as well as the convergence of REPP.
In what follows, we denote the diameter of set a A M by |A| := sup{dist(x, y) : x, y A}, and for any x M we define the translation of A by x as the set A + x := {a + x : a A}.
Proof of Theorem D. We want to start by showing that the condition D2(un) can be deduced from the decay of correlations as in the deterministic case. From our assumption, the random dynamical system has (annealed) decay of correla-tions, i.e., there exists a Banach space C of real-valued functions such that for all C and L1(µ ).
Expanding and piecewise expanding maps on the circle with anfinite number of discontinuities
We give a general definition from [V97] of piecewise expanding maps on the circle which also includes the particular case of the continuous expanding maps: (1) there exist N0 and 0 = a0 < a1 < · · · < a = 1 = 0 = a0 for which the restriction of f to eachΞ i = (ai1 , ai) is of class C 1, with |Df (x)| > 0 for all x Ξi and i = 1, . . . , . In addition, for all i = 1, . . . , , gΞi = 1/|Df |Ξi | has bounded variation for i = 1, . . . , . We assume that (f |Ξi ) and gΞi admit continuous extensions toΞ i = [ai 1, ai], for each i = 1, . . . , . Since modifying the values of a map over a finite set of points does not change its statistical properties, we may assume that f is either left-continuous or right-continuous (or both) at ai, for each i = 1, . . . , (possibly for all i’s at the same time).
Then let P (1) be some partition of S 1 into intervals Ξ such thatΞ Ξi for some i and (f |Ξ) is continuous. Furthermore, for n 1, let P (n) be the partition of S 1 such that P (n)(x) = P (n)(y) if and only if P (1)(f j (x)) = P (1)(f j (y)) for all 0j < n. Given Ξ P (n), denote gΞ(n) = 1/|Df n|Ξ|;
(2) there exist constants C1 > 0, 1 < 1 such that sup gΞ(n)C1 1n for allΞ P (n) and all n1;
(3) for every subinterval J of S 1, there exists some n1 such that f n(J ) = S 1.
Table of contents :
Acknowledgements
Resumo
Abstract
R´esum´e
1 Introduction
2 Preliminaries
2.1 General setting
2.2 Extreme Value Theory
2.3 Rare Event Point Processes
2.4 Hitting/Return Time Statistics
3 Deterministic Dynamics
3.1 Statement of the main results
3.2 Twofold dichotomy for deterministic systems
3.3 The dichotomy for specific systems
3.3.1 Rychlik maps
3.3.2 Piecewise expanding maps in higher dimensions
3.4 The extremal behaviour at discontinuity points
4 Stochastic Dynamics
4.1 Extremes for stochastic dynamics: direct approach
4.1.1 Statement of the main results
4.1.2 Laws of rare events for stochastic dynamics
4.1.3 Expanding and piecewise expanding maps on the circle with a finite number of discontinuities
4.1.4 Expanding and piecewise expanding maps in higher dimensions
4.2 Extremes for stochastic dynamics: spectral approach
4.2.1 The setting
4.2.2 Limiting distributions
4.2.3 Checking assumptions (A1)-(A4)
4.2.4 Extremal index
Bibliography
