Markov Extensions, Zeta Functions, and Fredholm Theory for Piecewise Invertible Dynamical Systems

Abstract

Abstract. Transfer operators and zeta functions of piecewise monotonie and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.

Entrance time functions for flat spot maps

“…and If (γ ) ∈ (3/10, 1/3) then q(γ ) 13, so E n = E n (γ ) is an interval for 1 n 12, and , so lemma 6.2 implies that ω(γ ) is weakly to the right of E 1 . But ω(γ ) is weakly to the left of E 1 by lemma 6.3, so in fact ω(γ ) ∈ E 1 (γ ).…”

“…As mentioned in section 1, first entrance time functions have attracted the attention of various authors, though with less emphasis on their fine detail. Notably, the field of open dynamical systems is concerned with a privileged subset F (referred to as a hole [1,5,7,8,16] or a trap [13]) of a phase space of some dynamical system, and the escape (or extinction, cf [13]) of orbits into F . Following the pioneering work of Pianigiani and Yorke [16], the primary objects of attention are (absolutely continuous) conditionally invariant measures µ and the escape rate − lim n→∞…”

“…, where E m is defined as in notation 3.10 (for further details see [1,5,7,8,13,16] and references therein). For example if T is any member of the standard family of flat spot maps considered in section 4 onwards, and the hole F is its flat spot, then the corresponding escape rate equals 1/2, and normalized Lebesgue measure µ on T \ F is conditionally invariant with eigenvalue 1/2, i.e.…”

“…The time taken to reach F is the main focus of this paper: in theorem 3. 13 we give an explicit description of the first entrance time function e T , defined by e T (x) = inf{i 0 : T i (x) ∈ F } (cf figures 2, 3 and 5). Although first entrance time functions (for arbitrary maps T and subsets F of phase space) have been studied widely in the dynamical systems literature (see, e.g., [1,5,7,8,13,16] and remark 3.24), it is rarely feasible to obtain explicit descriptions of these functions.…”