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Dynamical Zeta Functions For SUnimodal Maps
, 1999
"... . Let f be a nonrenormalizable Sunimodal map. We prove that f is a ColletEckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map. 1. Introduction A unimodal map f : [0; 1] ! [0; 1] is called Sunimodal if f(0) = f(1) = 0 and if it has nonpositive Schwarzian derivat ..."
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. Let f be a nonrenormalizable Sunimodal map. We prove that f is a ColletEckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map. 1. Introduction A unimodal map f : [0; 1] ! [0; 1] is called Sunimodal if f(0) = f(1) = 0 and if it has nonpositive Schwarzian derivative Sf = f 000 f 0 \Gamma 3 2 ( f 00 f 0 ) 2 . For such a map set '(x) := log jf 0 (x)j and 'n (x) := '(x) + '(fx) + \Delta \Delta \Delta + '(f n\Gamma1 x). Let \Pi n = fx 2 [0; 1] : f n (x) = xg and define for t 2 R the zeta function i t (z) = exp 1 X n=1 z n n i n;t where i n;t = X x2\Pi n e (t\Gamma1)' n (x) : Observe that i 0 (z) is just the usual dynamical zeta function. Denote per := inffj(f n ) 0 (x)j 1=n : n ? 0; x 2 \Pi n g : Nowicki and Sands [6] proved that per ? 1 (i.e. f is uniformly hyperbolic on periodic orbits) if and only if f satisfies the ColletEckmann condition (i.e. there are C ? 0 and CE ? 1 such that j(f n ) 0 (fc)j C n ...
FULL LARGE DEVIATION PRINCIPLE FOR BENEDICKSCARLESON QUADRATIC MAPS: ACIP AS A REFERENCE
, 2011
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Absolutely Continuous Invariant Measures As Equilibrium States For Piecewise Invertible Maps
, 1999
"... We characterize, for piecewise invertible dynamical systems in arbitrary dimension, the invariant probability measures which are absolutely continuous w.r.t. a given measure m with jacobian J as the equilibrium states w.r.t. \Gamma log J . We assume mainly bounded distortion of log J and negativity ..."
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We characterize, for piecewise invertible dynamical systems in arbitrary dimension, the invariant probability measures which are absolutely continuous w.r.t. a given measure m with jacobian J as the equilibrium states w.r.t. \Gamma log J . We assume mainly bounded distortion of log J and negativity of the pressure of the boundary of the pieces of the map. The proof uses shadowing (defined in [5,7] for Markov extensions) to apply an abstract proposition of F. Ledrappier [27] (introduced to characterize a.c.i.m.'s on the interval) using topological pressure estimates. Contrarily to [15,24], we do not use Markov extensions or the more usual transfer operator techniques. We apply this first to Lebesgue measure and multidimensional piecewise expanding maps with Jacobian not necessarily Holder and satisfying a generic condition which even holds for all piecewise expanding and affine mappings in the plane. As a corollary, we get existence of finitely many ergodic a.c.i.m.'s. Second, we con...
EQUILIBRIUM STATES FOR NONUNIFORMLY EXPANDING MAPS: DECAY OF CORRELATIONS AND STRONG STABILITY
, 2012
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LARGE DEVIATION PRINCIPLE FOR UNIMODAL MAPS SATISFYING THE COLLETECKMANN CONDITION
"... large deviation principle, free energy. Let I be the compact interval [0, 1] of the real line R. We denote by M the space of the Borel probability measures on I equipped with the weak topology and by m Lebesgue measure. For a map f: I → I nonsingular with respect to m we say that the large deviatio ..."
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large deviation principle, free energy. Let I be the compact interval [0, 1] of the real line R. We denote by M the space of the Borel probability measures on I equipped with the weak topology and by m Lebesgue measure. For a map f: I → I nonsingular with respect to m we say that the large deviation principle holds if there is an upper semicontinuous function q: M → [−∞, 0], called the rate function, satisfying lim inf n→∞ 1 n logm {x ∈ I: δnx ∈ G} ≥ sup µ∈G q(µ) for each open set G ⊂ M, and lim sup n→∞ 1 n logm {x ∈ I: δnx ∈ C} ≤ max µ∈C q(µ) for each closed set C ⊂ M, respectively, where δnx:=
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"... Mycobacterium leprae intracellular survival relies on cholesterol accumulation in infected macrophages: a ..."
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Mycobacterium leprae intracellular survival relies on cholesterol accumulation in infected macrophages: a