Topological and statistical attractors for interval maps

Abstract

We use the concept of Baire Ergodicity and Ergodic Formalism introduced in [Pi21] to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the wandering intervals attractors. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise C 2 maps with discontinuities. For C 2 interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map has abundance of historical behavior.

Thermodynamic formalism for expanding measures

“…As a consequence, supp η ⊂ ∂(F (g)) and, as a free expanding potential is an expanding potential, it follows from item (5) and that η ∈ E(g) satisfies supp η = supp µ and it is the unique possible equilibrium state for ϕ. It is known that every continuous transitive interval map is strongly transitive (see for instance Proposition 4.10 of [49]) and h top (f ) > 0 (see for instance Corollary 4.6.11 in [5] or the original proof by Block and Coven [8]). As f is C 1+ , every ergodic invariant measure ν with h ν (f ) > 0 has positive Lyapunov exponent.…”