[Paper Review] Decay of correlations on towers with non-Holder continuous Jacobian and non-exponential return time

Jérôme Buzzi, Véronique Maume‐Deschamps|arXiv (Cornell University)|May 7, 2004

Mathematical Dynamics and Fractals13 references3 citations

TL;DR

This paper establishes subexponential upper bounds on the decay of correlations for tower systems with non-Hölder continuous Jacobians and subexponentially decaying return times. By extending cone contraction techniques, it shows the correlation decay rate is governed by the slower of the two decay mechanisms—variation of the Jacobian and return time statistics—providing a unified framework for systems with both smoothness and return time obstructions.

ABSTRACT

We establish upper bounds on the rate of decay of correlations of tower systems with summable variation of the Jacobian and integrable return time. That is, we consider situations in which the Jacobian is not Holder and the return time is only subexponentially decaying. We obtain a subexponential bound on the correlations, which is essentially the slowest of the decays of the variation of the Jacobian and of the return time.

Motivation & Objective

• To analyze the rate of decay of correlations in tower systems where both the Jacobian lacks Hölder continuity and the return times decay subexponentially.
• To overcome the limitations of prior works that treated only one of these two obstructions separately.
• To establish a unified upper bound on correlation decay that reflects the minimum of the two decay rates.
• To provide a general framework applicable to non-uniformly hyperbolic systems, including maps with indifferent fixed points.
• To extend the use of cone techniques to handle non-smooth Jacobians and non-exponential return time distributions simultaneously.

Proposed method

• Use of Birkhoff's cones and projective metrics to analyze the action of the transfer operator on function spaces.
• Construction of a sequence of cones $ C_j $ of functions with controlled variation, adapted to the dynamics and return time statistics.
• Definition of a contraction rate $ \gamma_j < 1 $ for the transfer operator after $ k_j $ iterations, ensuring convergence in the uniform norm.
• Derivation of the product $ u_n = \prod_{j=2}^{\ell(n)} \gamma_j $ as the main decay rate, with $ \ell(n) $ being the largest index such that $ k_1 + \cdots + k_{\ell(n)} \leq n $.
• Explicit estimation of $ u_n $ in three regimes: exponential, stretched exponential, and polynomial decay of $ \omega_n $ and $ \hat{\nu}(\Delta_n) $, using asymptotic analysis of $ k_j $ and $ \gamma_j $.
• Application of the method to a non-Hölder map with an indifferent fixed point, showing correlation decay in $ \mathcal{O}(n^{-\min(\alpha, 1/\gamma - \varepsilon) + 1}) $ for any $ \varepsilon > 0 $.

Experimental results

Research questions

• RQ1 What is the rate of decay of correlations in tower systems when both the Jacobian is non-Hölder and the return time distribution is subexponential?
• RQ2 Can cone methods be extended to handle the simultaneous presence of non-smoothness and heavy-tailed return times?
• RQ3 How do the decay rates of the Jacobian variation and return time statistics interact in determining the overall mixing rate?
• RQ4 Is it possible to derive a subexponential upper bound on correlation decay that reflects the slower of the two decay mechanisms?
• RQ5 What is the correlation decay rate for a non-Hölder interval map with an indifferent fixed point, as derived from the general framework?

Key findings

• The decay of correlations is bounded by $ \mathcal{O}(n^{-\min(\alpha, \beta - \varepsilon) + 1}) $ for any $ \varepsilon > 0 $, where $ \alpha $ controls the variation of the Jacobian and $ \beta $ the tail of the return time distribution.
• For polynomial decay of $ \omega_n = \mathcal{O}(n^{-\alpha}) $ and $ \hat{\nu}(\Delta_n) = \mathcal{O}(n^{-\beta}) $, the correlation decay is $ \mathcal{O}(n^{-\min(\alpha - 1, \beta - 1 - \varepsilon)}) $, with $ \varepsilon > 0 $ due to technical limitations in the method.
• In the stretched exponential case with $ \omega_n = \mathcal{O}(e^{-n^\alpha}) $ and $ \hat{\nu}(\Delta_n) = \mathcal{O}(e^{-n^\beta}) $, the decay is $ \mathcal{O}(e^{-n^{\min(\alpha, \beta) - \varepsilon}}) $, matching the slower of the two decay rates.
• The method achieves a contraction coefficient $ \gamma_j = \max(1/D, D^{-\gamma/(α-1)}) $ in the polynomial case, with the final decay rate depending on the relative sizes of $ \alpha $ and $ \beta $.
• The application to a non-Hölder map with $ f(x) $ having an indifferent fixed point yields a correlation decay rate of $ \mathcal{O}(n^{-\min(\alpha, 1/\gamma - \varepsilon) + 1}) $, which is the first such result for such systems.
• The bound depends only on the $ L^1 $-norm of the observable $ \psi $, which is crucial for studying asymptotic laws of return times in non-uniformly hyperbolic systems.