A non-uniform Bowen s equation and connections to multifractal analysis

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1 A non-uniform Bowen s equation and connections to multifractal analysis Vaughn Climenhaga Penn State November 1, 2009

2 1 Introduction and classical results Hausdorff dimension via local dimensional characteristics Conformal repellers 2 Non-compact sets Non-uniformly expanding maps 3 Lyapunov spectra The Birkhoff spectrum

3 Global dimensional characteristics Hausdorff dimension via local dimensional characteristics Conformal repellers Quantify a fractal set Z using various global dimensional quantities. Hausdorff dimension dim H Z: critical value of t for t-dimensional Hausdorff measure m(z, t), defined using covers by metric balls B(x, ε). Replace B(x, ε) with Bowen balls B(x, n, δ): obtain topological entropy h top Z as a dimensional characteristic (Bowen). Give the balls B(x, n, δ) different weights according to S n ϕ(x) (Birkhoff sum): obtain topological pressure P Z (ϕ) as a dimensional characteristic (Pesin Pitskel ). (The latter two quantities depend on the underlying system f.)

4 Local dimensional characteristics Hausdorff dimension via local dimensional characteristics Conformal repellers Given an invariant measure µ for f, we have analogous local quantities: pointwise dimension d µ, local entropy h µ, Lyapunov exponent λ. d µ (x) = lim ε 0 log µ(b(x, ε)) log ε µ(b(x, ε)) ε dµ(x) h µ (x) = lim lim 1 µ(b(x, n, δ)) δ 0 n n λ(x) = lim n µ(b(x, n, δ)) e nhµ(x) 1 n log (f n ) (x) B(x, n, δ) B(x, δe nλ(x) ) Where the limits exist, we expect to find log µ(b(x, ε)) log µ(b(x, n, δ)) d µ (x) = lim = lim = h µ(x) ε 0 log ε n nλ(x) λ(x)

5 From local to global Introduction Hausdorff dimension via local dimensional characteristics Conformal repellers Local quantities can be used to determine global quantities. If µ(z) > 0 and d µ (x) = t for µ-a.e. x Z, then dim H Z t. If d µ (x) = t for every x Z, then dim H Z = t. Brin Katok and Birkhoff: For µ ergodic, h µ (x) and λ(x) are constant µ-a.e. and equal to h(µ) and λ(µ). d µ (x) = h(µ) =: d(µ) λ(µ) µ-a.e. Not quite enough: we need a measure µ such that d µ (x) is constant everywhere. d µ (x) = h µ(x) λ(x) = t h µ(x) tλ(x) = 0

6 Hausdorff dimension via local dimensional characteristics Conformal repellers The Gibbs condition and Bowen s equation We want a measure µ such that lim n 1 n log(µ(b(x, n, δ)) t 1 n S n(log f )(x) = 0 for every x. Now we are in thermodynamics...gibbs condition for µ, ϕ: there exist M > 0 and P = P(ϕ) R such that 1 M µ(b(x, n, δ)) e np+snϕ(x) M for all x, n, δ. For every x, this implies lim 1 n n log(µ(b(x, n, δ)) + 1 n S nϕ(x) = P(ϕ). Thus we need P( t log f ) = 0: Bowen s equation. If this holds and µ is a Gibbs measure for t log f, then d µ (x) = t everywhere, so dim H C = t.

7 Conformal repellers: Bowen and Ruelle Hausdorff dimension via local dimensional characteristics Conformal repellers General classical setting: M a Riemannian manifold, V M open, f : V M conformal and C 1+ε. Suppose J M has the following properties: 1 J is compact. 2 J is maximal: J = {x V f n (x) V for all n}. 3 f is topologically mixing on J. 4 f is uniformly expanding on J. Theorem (Ruelle, 1982) Under these assumptions, there exists a unique t R such that P( t log Df ) = 0. This t is the Hausdorff dimension of J.

8 A multifractal decomposition Non-compact sets Non-uniformly expanding maps Given α R, consider K L α = {x λ(x) = α}. Let X = {x λ(x) does not exist}. Then ( ) J = X α R is a multifractal decomposition of the repeller J. Every ergodic measure µ is supported on some K L α. Observation: Let µ be the Gibbs measure for t log f, where t = dim H J, and let α = λ(µ). Then µ(k L α ) = 1, and so dim H K L α = dim H J. What about the level sets K L α for other values of α? What is their Hausdorff dimension? K L α

9 Beyond Gibbs measures Non-compact sets Non-uniformly expanding maps Goal: Use Bowen s equation to determine Hausdorff dimension of non-compact sets, such as K L α. Problem: The classical thermodynamic formalism says nothing about the existence of Gibbs measures on such sets. Solution: Barreira and Schmeling give a proof that does not involve measures. This removes the following hypotheses from Ruelle s result: compactness, maximality, topological mixing. The following hypotheses remain: conformality, uniform expansion. We must keep conformality or else deal with the fact that log Df n (x) is no longer a Birkhoff sum in the non-conformal case. Uniform expansion can be significantly weakened, provided we still have bounded distortion and asymptotic exponential expansion.

10 Beyond uniform expansion Non-compact sets Non-uniformly expanding maps Given a conformal map f : X X and a set Z X, consider the following hypotheses: 1 f has no critical points ( Df (x) is non-vanishing). 2 Every point x Z has tempered contraction: for every ε > 0 we have inf 0 k n< (log Df n k (f k (x)) + nε) >. (Unbounded contraction does not happen too quickly along an orbit: automatic if λ(x) = λ(x) or if Df (x) 1 everywhere.) 3 Every point x Z has 0 < λ(x) λ(x) <. Theorem (C., 2009) Under the above hypotheses, dim H Z = inf{t P Z ( t log Df ) 0} = sup{t P Z ( t log Df ) > 0}.

11 Application to multifractal analysis Lyapunov spectra The Birkhoff spectrum Let f : X X be conformal. Two multifractal spectra: Entropy spectrum for Lyapunov exponents: L E (α) = h top K L α Dimension spectrum for Lyapunov exponents: L D (α) = dim H K L α One can show that P K L α ( t log Df ) = h top K L α tα. For α > 0, K L α satisfies the second and third hypotheses of the theorem. Thus L D (α) = 1 α L E(α) provided f has no critical points. How do we get L E (α)? This takes us deeper into multifractal analysis...

12 Lyapunov spectra The Birkhoff spectrum Other multifractal spectra: the Birkhoff spectrum General scheme: quantify level sets of local quantity (λ(x), h µ (x), d µ (x)) using global dimensional quantity (dim H, h top ). Lyapunov spectrum is particular case of a more general spectrum. Given ϕ: X R, K B α = { x X The Birkhoff spectrum of ϕ is } 1 lim n n S nϕ(x) = α. B(α) = h top K B α. Guiding philosophy: Obtain B(α) as Legendre transform of T(q) = P(qϕ). Legendre transform acts on functions: Maps convex to concave and vice versa. Applying two Legendre transforms gives the convex/concave hull.

13 A general result Introduction Lyapunov spectra The Birkhoff spectrum Known results: Work with particular class of systems, use tools specific to that class to analyse first T(q) and then B(α). Theorem (C., 2009) Let X be a compact metric space, f : X X be continuous, and ϕ: X R be continuous. 1 T(q) = sup α R (B(α) + qα). 2 If T(q) is differentiable and equilibrium states exist for each q, then B(α) = inf (T(q) qα). q R In particular, B(α) is concave and differentiable. T non-differentiable at q: phase transition. We get partial results for the part of the spectrum away from the phase transition.

14 Examples Introduction Lyapunov spectra The Birkhoff spectrum Lyapunov spectra: Uniformly expanding conformal repeller (Weiss). Non-uniformly expanding interval maps: possible phase transitions (Pollicott Weiss, Nakaishi, Gelfert Rams). Birkhoff spectrum: (no requirement of conformality) Uniformly hyperbolic map, Hölder continuous potential (Pesin Weiss). Uniformly hyperbolic map, non-hölder potential: possible phase transitions (thermodynamics by Sarig, Hu, Pesin Zhang). Unimodal interval maps, Hölder continuous potentials with supϕ inf ϕ < h top (f ) (thermodynamics by Bruin Todd).

15 Phase transitions Introduction Lyapunov spectra The Birkhoff spectrum How to handle phase transitions? Approximation from within by subsystems with no phase transitions. Theorem (C., 2009) Let f, X, ϕ be as before. Suppose that there exists a sequence of compact f -invariant subsets X n X such that 1 q P Xn (qϕ) is differentiable for all q; 2 equilibrium states exist for all q; and 3 lim n P Xn (qϕ) = P X (qϕ) for all q. Then B(α) = inf q R (T(q) qα) for all α. This theorem applies to any potential ϕ on a uniformly hyperbolic map which is Hölder continuous except at some finite number of periodic points.