[Paper Review] The Central Limit Theorem for function systems on the circle

Tomasz Szarek, Anna Zdunik|arXiv (Cornell University)|Mar 30, 2017

Mathematical Dynamics and Fractals9 references3 citations

TL;DR

This paper establishes the Central Limit Theorem (CLT) for iterated function systems (IFS) on the circle under minimal assumptions, specifically the minimality of the semigroup action, without requiring additional regularity of the generating homeomorphisms. The proof leverages the Maxwell-Woodroofe condition for Markov chains and introduces the e-property to ensure ergodicity and asymptotic stability, extending CLT to arbitrary initial distributions and general probability vectors via rational approximation.

ABSTRACT

The Central Limit Theorem for Iterated Functions Systems on the circle is proved. We study also ergodicity of such systems.

Motivation & Objective

• To establish the Central Limit Theorem (CLT) for Lipschitz observables in iterated function systems (IFS) on the circle under minimal assumptions.
• To extend the CLT beyond systems with spectral gap, allowing arbitrary initial distributions and general probability vectors.
• To provide a framework for ergodicity and asymptotic stability of IFS using the e-property of Markov operators.
• To offer alternative proofs of known ergodic results using the e-property, applicable to IFS with minimal dynamical assumptions.

Proposed method

• Uses the Maxwell-Woodroofe condition as a sufficient criterion for CLT in stationary Markov chains generated by the IFS.
• Applies the e-property of Feller operators to ensure equicontinuity of iterated functions, which implies separation of ergodic invariant measures.
• Employs the Cesáro e-property to prove weak convergence of empirical measures to the invariant measure for any starting point in a full-measure open set.
• Constructs a symbolic extension with equal probabilities via rational approximation of general probability vectors, preserving the invariant measure and CLT behavior.
• Estimates the difference in characteristic functions between trajectories starting from different points using Lipschitz continuity and the e-property.
• Uses approximation by rational probability vectors to extend the CLT from equal to general probabilities, maintaining uniform bounds on convergence rates.

Experimental results

Research questions

• RQ1 Under what minimal conditions does the Central Limit Theorem hold for iterated function systems on the circle with Lipschitz observables?
• RQ2 Can the CLT be established for IFS without requiring spectral gap or additional regularity of the maps?
• RQ3 How can the e-property of Markov operators be used to prove ergodicity and asymptotic stability of IFS on the circle?
• RQ4 Is the CLT valid for general initial distributions and arbitrary probability vectors in the IFS framework?
• RQ5 Can the CLT be extended from equal to general probability vectors using approximation techniques?

Key findings

• The CLT holds for IFS on the circle with Lipschitz observables under the sole assumption of minimality of the semigroup action, without requiring smoothness of the maps.
• The Maxwell-Woodroofe condition is sufficient for CLT in the IFS setting, enabling the result for arbitrary initial distributions.
• The e-property ensures that different ergodic invariant measures have disjoint supports, which is key to proving uniqueness and stability.
• Asymptotic stability and ergodicity of the IFS follow from the Cesáro e-property under mild conditions, including existence of a full-measure open invariant set.
• The CLT extends to general probability vectors via rational approximation, with convergence uniform in the approximation parameter.
• The proof remains valid for Hölder continuous observables with minor modifications, indicating robustness of the framework.