Exponential mixing and almost sure limit theorems in dynamical systems

A goal of ergodic theory is to understand the stochastic behaviours of deterministic systems. Given a measure preserving system (𝑋, 𝑓, μ), decay of correlations ensures that for reasonable ω : 𝑋 → ℝ,(ω(𝑓ⁿ𝑥)), 𝑛∈N asymptotically behaves like an i.i.d. process and results analogous to classical probabilistic theorems for i.i.d. sequences can be proved. Decay rates for uniformly hyperbolic maps are often exponential whereas non-uniformly hyperbolic systems can have troublesome rates, e.g. subexponential or polynomial. A common approach to study decay of correlations is via the corresponding symbolic space, which admits the same rate of mixing. Since non-uniformly hyperbolic systems are often modelled by countable Markov shifts which are non-compact, it requires a more exhausting machinery to prove analogous statements for finite shifts. This thesis will first review some thermodynamic results for subshifts of finite types (SFT) and countable Markov shifts (CMS) then focus on CMS withstrong positive recurrence (SPR), a property shown to be equivalent to the spectral gap property, which guarantees exponential mixing rates and other desirable features. For CMS satisfying certain topological boundary conditions, we will show that SPR is characterised by the ergodic averages over periodic orbits. Examples are provided to demonstrate that our condition is rather weak. In Chapters 3 and 4, we prove two sets of almost sure results using the Borel-Cantelli lemmas for fast mixing systems. Firstly, we show that the asymptotics of the cover times are almost surely quantified by the Minkowski dimensions, which dictate the growth of hitting times to geometrically small sets. The second set of theorems shows that for a point in a topological Markov shift, the length of the longest matching substrings grows exponentially depending on the Rényi entropy of the Gibbs measure. Such quantitative results extend analogously to the shortest distance problem for interval maps.