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W-maps and harmonic W-maps and harmonic Contents averages averages Contents Contents Harmonic mean (average) Harmonic mean (average) Harmonic mean (average) W-maps and harmonic averages W-map W-map Acim Stability of map τ Acim Stability of map τ July 2012 - Barcelona W-map The results The results Stronger Lasota-Yorke Stronger Lasota-Yorke inequality inequality Acim Stability of map τ Paweł Góra 1 in collaboration with Zhenyang Li, Minimax problem Minimax problem Abraham Boyarsky, Harald Proppe and Peyman Eslami. Lower bound for the Lower bound for the The results densities densities References References Concordia University Stronger Lasota-Yorke inequality July 2012 Minimax problem Lower bound for the densities 1 pgora@mathstat.concordia.ca W-maps and harmonic W-maps and harmonic Thanks Harmonic mean averages averages Contents Contents Harmonic mean (average) Harmonic mean (average) W-map W-map Acim Stability of map τ Acim Stability of map τ The results The results Stronger Lasota-Yorke Stronger Lasota-Yorke inequality inequality I am grateful to the organizers for the invitation and giving 2 Minimax problem Minimax problem H ( a , b ) = me a chance to present my results. 1 a + 1 Lower bound for the Lower bound for the b densities densities References References

W-maps and harmonic Acim Stability of map τ W-maps and harmonic W-map averages averages Contents Contents Harmonic mean (average) Harmonic mean (average) W-map W-map Acim Stability of map τ Acim Stability of map τ acim = absolutely continuous invariant measure The results The results Stronger Lasota-Yorke Stronger Lasota-Yorke We consider τ 0 with acim µ 0 and a family of perturbations inequality inequality τ a with acim’s µ a such that τ a → τ 0 as a → 0, say in Minimax problem Minimax problem Skorokhod metric. Lower bound for the Lower bound for the densities densities We say, τ 0 is acim stable if µ a → µ 0 as a → 0, say in weak ∗ References References topology. Keller constructed perturbations such that his W-map was not acim stable under these perturbations. First considered by G. Keller (1994) with s 1 = s 4 = 4, s 2 = s 3 = 2. W-maps and harmonic W-maps and harmonic Our perturbations Why is there a problem? averages averages Contents Contents Harmonic mean (average) Harmonic mean (average) W-map W-map Acim Stability of map τ Acim Stability of map τ From standard Lasota-Yorke (1973) inequality it follows The results The results that τ is acim stable if | τ ′ | ≥ λ > 2. Stability of isolated Stronger Lasota-Yorke Stronger Lasota-Yorke inequality inequality eigenvalues and corresponding eigenfunctions of Minimax problem Minimax problem Frobenius-Perron operator was proved by Keller and Lower bound for the Lower bound for the Liverani (1999). densities densities Standard method to improve the slope is to consider an References References iterate of τ . It does not work for perturbations of a map with a turning fixed point.

Iterates of perturbed τ 0 I W-maps and harmonic W-maps and harmonic The results averages averages Contents Contents Three cases: Harmonic mean (average) Harmonic mean (average) Second iterates for a = 0 . 10 and a = 0 . 05: W-map s 2 + 1 1 W-map s 3 > 1: There exists a small invariant subinterval Acim Stability of map τ Acim Stability of map τ around the turning fixed point x 0 and The results The results Stronger Lasota-Yorke Stronger Lasota-Yorke µ a → δ { x 0 } , inequality inequality Minimax problem Minimax problem ∗ -weakly. Lower bound for the Lower bound for the densities densities s 2 + 1 1 References References s 3 = 1: for example s 2 = s 3 = 2. τ a are exact on [ 0 , 1 ] and µ a → αδ { x 0 } +( 1 − α ) µ 0 , ∗ -weakly. To prove this we used the general formulas for acim of piecewise linear eventually expanding maps (Góra, 2009). W-maps and harmonic W-maps and harmonic The results: Stronger Lasota-Yorke inequality: averages averages Contents Contents Theorem Harmonic mean (average) Harmonic mean (average) Let τ : [ 0 , 1 ] → [ 0 , 1 ] be piecewise expanding with q W-map W-map branches, piecewise C 1 + 1 and satisfy Acim Stability of map τ Acim Stability of map τ s 2 + 1 1 s 3 < 1: � 1 The results The results � + 1 τ 0 is acim stable, i.e., Stronger Lasota-Yorke η = max < 1 , (1) Stronger Lasota-Yorke inequality s i s i + 1 inequality 1 ≤ i < q Minimax problem Minimax problem µ a → µ 0 , where s i = min | τ ′ i | , i = 1 , 2 ,..., q. Lower bound for the Lower bound for the densities densities Then, for every f ∈ BV ([ 0 , 1 ]) , not only ∗ -weakly but also in L 1 . The proof is based on a References References slightly stronger Lasota-Yorke inequality (Eslami and Góra, � � � P τ f ≤ η f + γ I | f | dm . (2) to appear). I I γ = M 2 s 2 + 1 ≤ i ≤ q m ( I i ) , where s = min s i , I i is the domain of s min branch τ i and M is the common Lipschitz constant of τ ′ i , i = 1 , 2 ,..., q.

W-maps and harmonic W-maps and harmonic Stability A small detail averages averages Contents Contents The above inequality holds if we assume additionally that Harmonic mean (average) Harmonic mean (average) τ ( 0 ) , τ ( 1 ) ∈ { 0 , 1 } . This restriction can be removed W-map W-map considering our system onto a slightly bigger interval and Acim Stability of map τ Acim Stability of map τ properly extended: The results The results Now, the whole stability theory holds under the above Stronger Lasota-Yorke Stronger Lasota-Yorke slightly weaker assumption. inequality inequality In particular, Ulam’s approximation method works under Minimax problem Minimax problem the assumption (1), (Góra and Boyarsky, to appear in Lower bound for the Lower bound for the densities densities Discrete and Continuous Dynamical System - A). References References Ulam’s method works also for standard W-map ( s 1 = s 4 = 4, s 2 = s 3 = 2). W-maps and harmonic W-maps and harmonic Minimax Problem Proof: averages averages Contents Contents Proof: We have Harmonic mean (average) Harmonic mean (average) W-map W-map min α , β max { s 1 α , s 2 β } = min α max { s 1 α , s 2 ( c − α ) } . Acim Stability of map τ Acim Stability of map τ The constant � 1 � + 1 The results The results η = Stronger Lasota-Yorke The line f ( α ) = s 1 α is increasing while the line Stronger Lasota-Yorke s 1 s 2 inequality inequality g ( α ) = s 2 ( c − α ) is decreasing. The shows up in the following minimax problem: Minimax problem Minimax problem min α max { s 1 α , s 2 ( c − α ) } occurs where the lines intersect, Let s 1 , s 2 > 1 and α + β = c , where α , β > 0. Then, Lower bound for the Lower bound for the densities densities i.e., at s 2 c References References 1 α = , α , β max { s 1 α , s 2 β } = c . min s 1 + s 2 s 1 + 1 1 s 2 which gives α , β max { s 1 α , s 2 β } = s 1 s 2 c 1 = c . min s 1 + s 2 s 1 + 1 1 s 2