Limit theorems for non-hyperbolic automorphisms of the torus

Central limit theorem on hyperbolic groups

“…This method has been introduced in [24] and [23] and has been often used since then; see, for instance, [10,35]. See also [16] and [11], Appendix, for a survey of this method and [11], Section 2.4, for the use of this method in order to prove a CLT and an invariance principle in the context of products of independent random matrices.…”

Central limit theorem for linear groups

“…We will follow Gordin's method. This method has been introduced in [24] and [23] and has been often used since then; see, for instance, [10,35]. See also [16] and [11], Appendix, for a survey of this method and [11], Section 2.4, for the use of this method in order to prove a CLT and an invariance principle in the context of products of independent random matrices.…”

Exponential mixing of nilmanifold automorphisms

“…5] usually allows to deduce the proof of the central limit theorem from quantitative equidistribution of unstable foliations. Using this approach, the central limit theorem and its generalisations have been established for ergodic toral automorphisms in [19,17] and for ergodic automorphisms of 3-dimensional nilmanifolds in [4]. Here we extend these results to general nilmanifolds.…”

“…However, unless they are hyperbolic, the toral automorphisms lack the specification property and, in particular, don't have Markov partitions [20]. Nonetheless, it is known that ergodic toral automorphisms satisfy the central limit theorem and its refinements [19,17]. Regarding the quantitative aspects Lind established exponential mixing for ergodic toral automorphisms using Fourier analysis [21].…”