Rigorous pointwise approximations for invariant densities of non-uniformly expanding maps

Transcription

1 Ergod. Th. & Dynam. Sys. 5, 35, 8 44 c Cambridge University Press, 4 doi:.7/etds.3.9 Rigorous pointwise approximations or invariant densities o non-uniormly expanding maps WAEL BAHSOUN, CHRISTOPHER BOSE and YUEJIAO DUAN Department o Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE 3TU, UK W.Bahsoun@lboro.ac.uk, Y.Duan@lboro.ac.uk Department o Mathematics and Statistics, University o Victoria, PO Box 345 STN CSC, Victoria, BC, V8W 3R4, Canada cbose@uvic.ca Received 6 January 3 and accepted in revised orm 3 August 3 Abstract. We use an Ulam-type discretization scheme to provide pointwise approximations or invariant densities o interval maps with a neutral ixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate C ln m/m, where C is a computable ixed constant and m is the mesh size o the discretization.. Introduction Ulam-type discretization schemes provide rigorous approximations or dynamical invariants. Moreover, such discretizations are easily implementable on a computer. In [8] it was shown that the original Ulam method [3] is remarkably successul in approximating isolated spectrum o transer operators associated with piecewise expanding maps o the interval. In particular, it was shown that this method provides rigorous approximations in the L -norm or invariant densities o Lasota Yorke maps see [8] and reerences therein. This method has been also successul when dealing with multi-dimensional piecewise expanding maps [], and partially successul in providing rigorous approximations or certain uniormly hyperbolic systems [, ]. Recently, Blank [5] and Murray [] independently succeeded in applying the pure Ulam method in a non-uniormly hyperbolic setting. They obtained approximations in the L -norm or invariant densities o certain non-uniormly expanding maps o the interval. Although L approximations provide signiicant inormation about the long-term statistics o the underlying system, they are not helpul when dealing with rare events See [6] or examples where the pure Ulam method provides ake spectra or certain hyperbolic systems. In [], in addition to proving convergence, Murray also obtained an upper bound on the rate o convergence.

2 Pointwise approximations o invariant densities 9 in dynamical systems. In act, when studying rare events in dynamical systems [, 5] one oten obtains probabilistic laws that depend on pointwise inormation rom the invariant density o the system. In particular, extreme value laws o interval maps with a neutral ixed point depend pointwise on the invariant density o the map [3]. Statistical properties o non-uniormly expanding maps were studied by Pianigiani [], who irst proved existence o invariant densities o such maps. Later, it was independently proved in [4, 9, 4] that such maps exhibit polynomial decay o correlations. The slow mixing behaviour that such maps exhibit has made them good testing tools or real and diicult physical problems. The diiculty in obtaining pointwise approximations or invariant densities o interval maps with a neutral ixed point is twoold. Firstly, the transer operator associated with such maps does not have a spectral gap in a classical Banach space. Thereore, powerul perturbation results [6] are not directly available in this setting. Secondly, invariant densities o such maps are not L unctions. Consequently, to provide pointwise approximation o such densities, one should irst measure the approximations in a properly weighted L -norm. In this paper we use a piecewise linear Ulam-type discretization scheme to provide pointwise approximations or invariant densities o non-uniormly expanding interval maps. Our main result is stated in Corollary 3.. For x, ] we prove that the approximate invariant density converges pointwise to the true density at a rate C /x +α ln m/m, where C is a computable ixed constant, α, is a ixed constant, and m is the mesh size o the discretization. To overcome the spectral diiculties and the unboundedness o the densities which we discussed above, we irst induce the map and obtain a uniormly piecewise, expanding and onto map. Then we perorm our discretization on the induced space. Ater that we pull back, both the invariant density and the approximate one to the ull space and measure their dierence in a weighted L - norm. Full details o our strategy are given in 3.. In, we recall results on uniormly piecewise expanding and onto maps. Moreover, we introduce our discretization scheme and recall results about uniorm approximations or invariant densities o uniormly piecewise expanding and onto maps. In 3, we introduce our non-uniormly expanding system, set up our strategy, and state our main results, Theorem 3. and Corollary 3.. Section 4 contains technical lemmas and the proo o Theorem 3.. Section 5 presents an algorithm based on the result o Corollary 3. and discusses its easibility.. Preliminaries.. A piecewise expanding system. Let, B, ˆλ denote the measure space where is a closed interval, B is Borel σ -algebra and ˆλ is normalized Lebesgue measure on. Let ˆT : be a measurable transormation. We assume that there exists a countable partition P o, which consists o a sequence o intervals, P = {I i } i=, such that: or each i =,...,, ˆT i := ˆT is monotone, C, and it extends to a C unction I i on Ī i ; See also [] or another perturbation result.

3 3 W. Bahsoun et al ˆT i I i =, i.e. or each i =,...,, ˆT i is onto; 3 there exists a constant D > such that sup i sup x Ii ˆT x / ˆT x D; 4 there exists a number γ such that / ˆT i Let ˆL : L L denote the transer operator Perron Frobenius [4, 8] associated to ˆT : ˆL x = y= ˆT x y ˆT y. Under the above assumptions, among other ergodic properties, it is well known that see, or instance, [7] ˆT admits a unique invariant density ˆ, i.e. ˆL ˆ = ˆ. Moreover, ˆL admits a spectral gap when acting on the space o Lipschitz continuous unctions over []. We will denote by BV the space o unctions o bounded variation deined on the interval. Set BV := V +,, where V denotes the one-dimensional variation over. Then it is well known that BV, BV is a Banach space and ˆL satisies the ollowing inequality see, or instance, []: there exists a constant C LY > such that, or any BV, we have Inequality. is called the Lasota Yorke inequality. V ˆL γ V + C LY,.... Markov discretization. We now introduce a discretization scheme which enables us to obtain rigorous uniorm approximation o ˆ, the invariant density o ˆT. We use a piecewise linear approximation which was introduced by Ding and Li [9]. Let η = {c i } i= m be a partition o into intervals. Since uniorm partitions are the irst choice or numerical work, we set c i c i = /m, where is the length o. Everything we do can be easily modiied or non-uniorm partitions with only minor notational changes. Let ϕ i = χ [ci,c i ] and φ i x = m Let ψ i denote a set o hat unctions over η: x ϕ i dλ. ψ := φ, ψ m := φ m and or i =,..., m, ψ i := φ i φ i+.. For L, we set I i := [c i, c i ] and i := m I i dx, i =,,..., m, the average o over the associated partition cell. For L we set m Q m := ψ + i= i + i+ ψ i + m ψ m. Obviously, the operator Q m retains good stochastic properties, i.e.: or, Q m ; Qm =. In [], a Lasota Yorke inequality was obtained or Markov interval maps with a inite partition. The proo carries over or piecewise onto maps with a countable number o branches satisying the assumptions o..

4 Pointwise approximations o invariant densities 3 We now deine a piecewise linear Markov discretization o ˆL by P m := Q m ˆL..3 Notice that P m is a inite-rank Markov operator whose range is contained in the space o continuous, piecewise linear unctions with respect to η. The matrix representation o P m restricted to this inite-dimensional space and with respect to the basis {ψ i } is a row stochastic matrix. By the Perron Frobenius theorem or stochastic matrices [7], P m has a let invariant density ˆ m, i.e. ˆ m = ˆ m P m. The ollowing theorem was proved in []. THEOREM.. There exists a computable constant Ĉ such that, or any m N, ˆ ˆ m Ĉ ln m m. Remark.. We recall that in [] it was shown that the constant Ĉ, which is independent o m, can be computed explicitly. 3. Pointwise approximations or invariant densities o maps with a neutral ixed point 3.. The non-uniormly expanding system. Let I = [, ] be the unit interval, λ be the Lebesgue measure on [, ]. Let T : I I be a piecewise smooth map with two branches. We assume that: T = and there is an x, such that T = T [,x, T = T [x,] and T : [, x onto onto [,, T : [x, ] [, ]; T is C on [, x ], T is C on, x ] and T is C on [x, ]; T = and T x > or x, x ; T x > or x x, ; T and T have the orm T x = x + x +α + x +α δ x, T x = + + αxα + x α δ x, where, < α < and δ i x as x or i =, with δ x. It is well known that T admits a unique invariant density [4, 9,, 4] and the system I, T, λ exhibits a polynomial mixing rate [4, 9, 4]. Moreover, it is well known [4, 9, 4] that the T -invariant density,, is not an L -unction. In particular, near x =, x behaves like x α. Despite this diiculty, we will show that, or any x, ], one can obtain rigorous pointwise approximation o x. 3.. Strategy and the statement o the main result. Recall that α,. We irst deine a suitable Banach space that contains. More precisely, let B denote the set o continuous unctions on, ] with the norm B = sup x,] x +α x.

5 3 W. Bahsoun et al When equipped with the norm B, B is a Banach space. The act that B ollows rom [4, Lemma 3.3]. Our strategy or obtaining a pointwise approximation o consists o the ollowing steps. We irst induce T on I and obtain a ˆT which satisies the assumptions o.. On, we use Theorem. to say that ˆ m, the invariant density o the discretized operator P m := Q m ˆL, deined in equation.3, provides a uniorm approximation o ˆ the ˆT -invariant density. 3 Next we write in terms o ˆ, and deine a unction m as the pullback o ˆ m. 4 Finally, we use steps and 3 to prove that m B C ln m/m, and deduce a pointwise approximation o The induced system We induce T on := [x, ]. For n we deine Set For n, we deine x n+ = T x n. W := x, and W n := x n, x n, n. Z n := T W n. Then we deine the induced map ˆT : by Observe that ˆT x = T n x or x Z n. 3. T Z n = W n and τ Zn = n, where τ Zn is the irst return time o Z n to. An example o the map T and its induced counterpart ˆT are shown in Figures and respectively. It is well known see, or instance, [4] that the ˆT deined in 3. satisies the assumptions o., and, by Theorem., one can obtain a rigorous uniorm approximation o its invariant density ˆ. Moreover, by [3, Lemma 3.3],, the invariant density o T, can be written in terms o ˆ: c τ ˆx or x, x = ˆT T n x 3. c τ n= DT n T T n or x I \, x where ˆ is the ˆT -invariant density, cτ = k= τ Zk ˆµZ k, and ˆµ = ˆ ˆλ The approximate density and the statement o the main result. Set c τ,m ˆ m x or x, m x := de ˆ m T T n x c τ,m n= DT n T T n or x I \, x 3.3 In what ollows, we only use the metric properties o B. In particular, the completeness o B is not needed in our proos.

6 Pointwise approximations o invariant densities x FIGURE. A typical example o a map T which belongs to the amily deined in x 3 x x x z n z 3 z z FIGURE. This igure shows the induced map ˆT corresponding to the map T o Figure.

7 34 W. Bahsoun et al where ˆ m = P m ˆ m, and P m is the Markov discretization o ˆL deined in.3, cτ,m = k= τ Zk ˆµ m Z k, and ˆµ m = ˆ m ˆλ. The next result shows that the unction m deined in 3.3 provides a rigorous pointwise approximation o. THEOREM 3.. For any m N we have where m B C ln m m, C = Ĉ + x+α + M + α C 4 ; in particular, Ĉ is the computable constant o Theorem., and C := α + α M := C+α C 4 := + C 3 CLY γ + e C C α [ + δ x + δ x ], C := [ /α ] /α,, C 3 := + C α + α α C = x x +α +/α [ /α ] +/α. As a direct consequence o Theorem 3. we obtain the required pointwise approximation o. COROLLARY 3.. For any x, ] we have,, Proo. For x, ], we have x m x C x +α ln m m. x m x = x +α x+α x m x x +α m B ln m C x+α m. 4. Proos 4.. Technical lemmas. We irst introduce notation o certain unctions which appear in the proo o Theorem 3.. For x I \, set gx := T x/x +α T x, G x := x +α T T x and or n, G n x := x +α DT n T T n x.

8 LEMMA 4.. For x I \, we have [ + x α + x α δ x] +α + + α[x α + x α δ x] + Proo. Let and φ x := [ + x α + x α δ x] +α φ x := + + α[x α + x α δ x] + α + α [x α + x α δ x]. α + α [x α + x α δ x]. Note that φ = φ =. Thereore, to prove the lemma, it is enough to prove that φ x φ x. We have φ x = + α + ξxα ξ x, φ x = + α + αξxξ x, where ξx := x α + x α δ x. Notice that ξ x. Thus, we only need to show that + ξx α + αξx. 4. Indeed, 4. holds because + ξ α = + αξ = and [ + ξx α ] = α + ξx α ξ x αξ x = [ + αξx]. LEMMA 4.. For x I \, we have gx + C x α, where C = Proo. Using Lemma 4., we have Pointwise approximations o invariant densities 35 α + α [ + δ x + δ x ]. gx = T x/x +α T x = [ + xα + xα δ x]+α + + αx α + x α δ x + + α[xα + x α δ x] + α + α/[x α + x α δ x] + + αx α + x α δ x = + + α[xα + x α δ x] + + αx α + x α δ x + α + α/[xα + x α δ x] + + αx α + x α δ x α + α + [x α + x α δ x] α + α = + + δ x + δ xxα + C x α. LEMMA 4.3. Let x n = T n x. For n, we have x n C n /α, where C = [ /α ] /α. It is obvious that ξ = and or x >, ξx >.

9 36 W. Bahsoun et al Proo. Observe that C > T x = x. Thereore, the lemma is true or n =. Next, or n, we suppose that x n C n /α, and prove that x n C n /α. I it is alse, that is x n > C n /α, then by our inductive statement on x n, we have C n /α x n = T x n > C n /α [ + C α n + C α n δ C n /α ]. This is equivalent to [ n + n /α ] > C α [ + δ C n /α ]. By convexity o the unction z /α, n n [/α ] > C α [ + δ C n /α ], that is, C α < n n [/α ]/[ + δ C n /α ] < [ /α ] = C α. This represents a contradiction, and thereore, x n C n /α. This completes the proo o the lemma. LEMMA 4.4. For x I \, we have and or n, where M = C +α e C C α /. Proo. For n =, it is easy to see that For n, we have G n x = = = = x +α DT n T T n x G x x+α, G n x Mn +/α, G x x+α. x +α DT T T T T T n x T T x T x/t x +α T T T x T T n x +α T T n x x T x +α T n T x/t x +α T T x x T T T n x T n x/t n x +α T n T x

10 Pointwise approximations o invariant densities 37 = gt x gt x gt n x T T n x +α T T n x gt x gt x gt n n T x +α x. 4. By Lemmas 4. and 4.3, or any k, x [, x, we have gt k x + C T k x α + C T k x α Thereore, using 4. and 4.3, or n, we obtain = + C x k α + C C α k. 4.3 G n x n k= n gt k T x +α x n + C C α k C+α n +/α k= { n } = exp ln + C C α k C+α n +/α k= { n } exp C C α k C+α n +/α k= { exp C C α } C+α n +/α n Mn +/α. Finally we estimate the dierence between the normalizing constants appearing in equations 3. and 3.3. LEMMA 4.5. where and Proo. By Lemma 4.3, we have n ˆλZ n C 3, n= C 3 = + C α + α x α C = x x +α +/α [ /α ] +/α. λw n = x n x n = T x n x n = x x +α xn +α x x +α C +α n +/α = C n +/α.

11 38 W. Bahsoun et al Since T Z n = W n, we have n λz n λz + n= x = x = x x x = x n= n= n= n= n= + C + C n λw n nx n x n n + x n x n nx n x n + C n /α + + x /α dx α + α α n= x n x n C n +/α n= + C = x C 3. + x +/α dx This completes the proo o the lemma since ˆλ = λ / x. LEMMA 4.6. We have c τ,m c τ C 3 Ĉ ln m m. Proo. Using the act that c τ, c τ,m and Theorem., we have c τ,m c τ k= τ Zk ˆµ m Z k k= τ Zk ˆµZ k k= = k[ ˆµZ k ˆµ m Z k ] k= τ Zk ˆµ m Z k k= τ Zk ˆµZ k k ˆ ˆ m d ˆλ Z k k= ˆ ˆ m k ˆλZ k k= Ĉ ln m m C 3. In the last estimate, we have used Lemma 4.5. We now have all our tools ready to prove Theorem 3..

12 Proo o Theorem 3.. Using 3. and 3.3, we have m B = sup x +α x m x x,] sup x I \ x +α x m x + sup x +α x m x x = sup x +α x I \ n= DT n T T n x c τ ˆ T T n x c τ,m ˆ m T T n x + sup x +α c τ ˆx c τ,m ˆ m x. 4.4 x Notice that or x I \, and n, z n := T x. Then using the act that c τ, c τ,m, Theorem., Lemma 4.6, and 4.4, we obtain m B sup x +α x I \ n= DT n T T n x sup c τ ˆz n c τ,m ˆ m z n z n + sup c τ ˆx c τ,m ˆ m x x x I \ n= T n sup x +α DT n T T n x sup ˆz n ˆ m z n + c τ c τ,m sup ˆz n z n z n + sup x Ĉ ln m m Pointwise approximations o invariant densities 39 ˆ x ˆ m x + c τ c τ,m sup sup x I \ n= x ˆ x G n x +C 3 sup ˆz n ++C 3 sup ˆx z n x. 4.5 Since ˆ BV, we have sup x ˆx V ˆ + / x ˆ,. Thereore, using the Lasota Yorke inequality., we obtain sup x ˆx C LY / γ + / x. Using Lemma 4.4 and 4.5, we obtain m B C 4 Ĉ ln m + x+α m + Mn +/α n= = C 4 Ĉ ln m + x+α m + M n +/α n= C 4 Ĉ + x+α + M + α ln m m. 5. Algorithm and easibility Given a map T satisying the conditions o 3., and x, ], we provide an algorithm based on Corollary 3. that can be used to approximate x, the T -invariant density at the point x, up to a pre-speciied approximation error R.

13 4 W. Bahsoun et al 5.. Algorithm and output. Compute the constants, M, C 4 which appear in Theorem 3.. Compute an upper bound on the constant Ĉ which appears in Theorem.. 3 Then use, to compute C which appears in Theorem Find m, number o bins, such that ln m m x +α C R 3. 5 Compute an approximate ixed point., where ε is chosen such that ˆ m > o P m so that ˆ m ˆ m ε on where m x = c τ,m c τ,m := k= k ˆµ m Z k and ˆµ = ˆ m ˆλ. 6 Find N such that m x m x R 3, ˆ m T T n x n= DT n T T n x c τ,mn N ˆ m T T n x n= DT n T T n m x x R 3, where c τ,m N := N k= k ˆµ m Z k. 7 The approximate value o x is given by :, m,n x := c τ,m N N ˆ m T T n x n= DT n T T n. x 5.. Feasibility. For, once the map T is given, the constants, M, C 4 can be computed analytically. For, the constant Ĉ appears in the approximation done on the induced system see Theorem.. The induced system is a uniormly expanding map. The computation o Ĉ can be done ollowing the ideas o [], which is based on the spectral stability result o [6]. 3 is a consequence o and. Recall that ˆ m is the ixed point o the inite rank operator P m deined in.3. Here ˆ m is the computer approximation o ˆ m, i.e. 5 o the Algorithm takes care o the computer roundo errors in computing the ixed point o P m. Since ˆ m >, we also ask in this computation that ˆ m >. Note that the strict positivity o ˆ m ollows rom the act that the induced map ˆT is a piecewise onto map, which implies that the matrix representation o P is irreducible, and consequently its Perron eigenvector is strictly positive see Perron Frobenius theorem [7]. It is very important to notice that the approximation m,n x is a inite sum.

14 Once C is computed in 3, with m := m we know, rom Corollary 3., that m x x R/3. For 5 we should ind out how small ε should be to ensure m x m x R 3. We propose the ollowing method. To work out explicitly all the constants needed in veriying 5, we suppose x W k, and T x = x + α x +α. In the ollowing estimates 5., 5.3 and 5.4, we prepare the ingredients to achieve our job. Firstly, ollowing exactly the argument o [3, Lemma 5.], or x W k, where and η k = DT n T T n x dk + k + + d, d = d = + αα Consequently, η k >, and or x W k, n= [ n + k k + α /α + α /α. DT n T T n x ηk, 5. ] α + α /α + d >, k + k ηk k η k. 5. Secondly, using the same argument as that in the proo o Lemma 4.6, we have c τ,m c τ,m C 3 ˆ m ˆ m. 5.3 Thirdly, it is well known that or g BV we have V Q m g g, where Q m is the discretization deined in.. Thereore, P m satisies the same Lasota Yorke inequality. as ˆL. In particular, this implies that ˆ m, the P m ixed point, satisies: ˆ m C LY γ Consequently, using and that c τ,m, we obtain m x m x ˆ m ˆ m k + Thus, to ensure m x m x R/3, that satisies k ηk [ k CLY + C 3 η k γ + ]. 5.5 ˆ m should be computed to a precision ˆ m ˆ m /k + /k η k k/η k [ + C 3 C LY / γ + / ] See, or instance, [9, Lemma.3] Pointwise approximations o invariant densities 4 R 3.

15 4 W. Bahsoun et al For 6, as in 5, we also suppose x W k, and T x = x + α x +α. Firstly, by 5., each term in the sum ˆ m T T n x n= DT n T T n 5.6 x is bounded above by ˆ m y k + ηk. n + k sup y sup y Since η k >, the tail o the sum, starting rom N +, in 5.6 can be approximated as ollows. Choose N such that ˆ m y N + k η k+ k + η k R η k Secondly, using k= k ˆµ m Z k, and recalling that α, and in y ˆ m y >, we have c τ,m c τ,m N = N + k ˆµ m Z k k= k ˆµ m Z k N k= k ˆµ m Z k sup y sup y in y Thirdly, using 5. and 5.8, we have N ˆ + k ˆλZ k m N k= k ˆµ m Z k N ˆ + C / k k /α m ˆ m ˆλZ sup y c τ,m c τ,m N m,n x in y in y sup y sup y ˆ m ˆ m ˆλZ C in y ˆ m ˆ m ˆλZ C N /α+ /α /α+ N /α. 5.8 m x ˆ m C ˆ m ˆλZ N /α+ k + ηk k /α k η k. 5.9 Choose N such that the right-hand side o 5.9 is not greater than R/6. Finally, choose N = max{n, N }. This will lead to the desired estimate m x m,n x R/3. In our analysis we veriied many items o the Algorithm analytically. Many o these steps may be veriied iteratively using a computer. For instance, N may be ound iteratively using a computer. Simply keep N k= k ˆµ m Z k in the denominator on the right-hand side o 5.9 instead o replacing it by in y ˆ m ˆλZ. The iterative method may ind a smaller value o N that achieves the job.

16 For 7, using the above six steps, we obtain x m,n x x m x + m x m x + m x m,n x R; i.e. the inite sum Pointwise approximations o invariant densities 43 m,n x := c τ,m N N ˆ m T T n x n= DT n T T n x is a rigorous approximation o x up to the pre-speciied error R. Acknowledgement. The authors would like to thank an anonymous reeree or comments that improved both the presentation and the content o the paper. REFERENCES [] H. Aytac, J. Freitas and S. Vaienti. Laws o rare events or deterministic and random dynamical systems. Preprint,, arxiv.org/pd/ [] W. Bahsoun and C. Bose. Invariant densities and escape rates: rigorous and computable approximations in the L -norm. Nonlinear Anal. 74, [3] W. Bahsoun and S. Vaienti. Metastability o certain intermittent maps. Nonlinearity 5, 7 4. [4] V. Baladi. Positive Transer Operators and Decay o Correlations Advanced Series in Nonlinear Dynamics, 6. World Science Publications, River Edge, NJ,. [5] M. Blank. Finite rank approximations o expanding maps with neutral singularities. Discrete Contin. Dyn. Syst. 3 8, [6] M. Blank, G. Keller and C. Liverani. Ruelle Perron Frobenius spectrum or Anosov maps. Nonlinearity 56, [7] C. Bose. Invariant measures and equilibrium states or piecewise C +α endomorphisms o the unit interval. Trans. Amer. Math. Soc , 5 5. [8] A. Boyarsky and P. Góra. Laws o Chaos: Invariant Measures and Dynamical Systems in one Dimension. Birkhäuser, Boston, 997. [9] J. Ding and T. A. Li. Convergence rate analysis or Markov approximations to a class o Frobenius Perron operators. Nonlinear Anal , [] G. Froyland. Finite approximation o Sinai Bowen Ruelle measures or Anosov systems in two dimensions. Random Comput. Dynam , [] G. Froyland. On Ulam approximation o the isolated spectrum and eigenunctions o hyperbolic maps. Discrete Contin. Dyn. Syst. 73 7, [] S. Galatolo and I. Nisoli. An elementary approach to rigorous approximation o invariant measures. Preprint,, arxiv:9.34. [3] M. Holland, M. Nicol and A. Török. Extreme value theory or non-uniormly expanding dynamical systems. Trans. Amer. Math. Soc. 364, [4] H. Hu. Decay o correlations or piecewise smooth maps with indierent ixed points. Ergod. Th. & Dynam. Sys. 4 4, [5] G. Keller. Rare events, exponential hitting times and extremal indices via spectral perturbation. Dyn. Syst. 7, 7. [6] G. Keller and C. Liverani. Stability o the spectrum or transer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci , 4 5. [7] P. Lancaster and M. Tismenetsky. The Theory o Matrices. Academic Press, Orlando, FL, 985. [8] C. Liverani. Rigorous numerical investigation o the statistical properties o piecewise expanding maps. A easibility study. Nonlinearity 43, [9] C. Liverani, B. Saussol and S. Vaienti. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys ,

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