Long periodic orbits of the triangle map

Source abstract

Let τ : [ 0 , 1 ] → [ 0 , 1 ] \tau :[0,1] \to [0,1] be defined by τ ( x ) = 2 x \tau (x) = 2x on [ 0 , 1 / 2 ] [0,1/2] and τ ( x ) = 2 ( 1 − x ) \tau (x) = 2(1 - x) on [ 1 / 2 , 1 ] [1/2,1] . We consider τ \tau restricted to the domain D N = { 2 a / p N , N ⩾ 1 , 0 ⩽ 2 a ⩽ p N , ( a , p ) = 1 } {D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\} where p p is any odd prime. Let k ⩾ 1 k \geqslant 1 be the minimum integer such that p N | 2 k ± 1 {p^N}|{2^k} \pm 1 . Then there are ( ( p − 1 ) ⋅ p N − 1 ) / 2 k (({\text {p}} - 1) \cdot {p^{N - 1}})/2k periodic orbits of τ | D N \tau {|_{{D_N}}} , having equal length, and there are k k points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval ( c , d ) (c,d) approaches d − c d - c as p N − 1 → ∞ {p^{N - 1}} \to \infty . An application to the irreducibility of certain nonnegative matrices is given.