Total Variation of a Curve Under Chaos on the Real Line and on a Finite Graph.

We show that if f : R → R is a continuous transitive map and γ : [ 0 , 1 ] → R is a nonconstant curve having finite total variation, then the total variation of f n ∘ γ tends to infinity exponentially as n → ∞ . A similar result is also proved for a Devaney chaotic map on a graph G. [ABSTRACT FROM AUTHOR]