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.

Large deviations in non-uniformly hyperbolic dynamical systems

“…However, we know of no suitable reference(s) that contain all the needed results in our particular setup. We will follow closely parts of [21,28,2], noting that most of the arguments in fact date back earlier.…”

Entrance time functions for flat spot maps

“…But f γ is an increasing function, so (14) and (15) imply that its unique zero ω(γ) lies in E i l (γ), as required. If A and A are any two subsets of an interval J, we say that A is weakly to the left (respectively the right) of A if there exists a ∈ A such that a ≤ a for all a ∈ A (respectively a ≤ a for all a ∈ A).…”

“…it is connected ). [17], the primary objects of attention are (absolutely continuous) conditionally invariant measures µ and the escape rate − lim n→∞ 1 n log µ (∪ m>n E m ), where E m is defined as in Notation 3.10 (for further details see [1,5,7,8,14,17] 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 normalised Lebesgue measure µ on T \ F is conditionally invariant with eigenvalue 1/2, i.e.…”

“…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. [1,5,7,8,14,17] and Remark 3.24), it is rarely feasible to obtain explicit descriptions of these functions. The standard family of flat spot maps T γ (see e.g.…”

Almost isomorphism for countable state Markov shifts

“…In a companion paper [4], we describe a reasonable class of Borel potentials which behaves well under almost isomorphism. We are also motivated by the use of countable state Markov shifts to code some partially or piecewise hyperbolic systems ( [5,8,9,25,26,27,29,30,56,57,59,64] and successors of [64]); investigations of coding relations among Markov shifts (see [12,13,14,15,16] and their references); and a longstanding finite state coding problem for Markov measures not of maximal entropy (8.4). This paper is dedicated to Klaus Schmidt and Peter Walters, on the occasion of their sixtieth birthdays.…”