Projective Cones for Sequential Dispersing Billiards

Periodic Lorentz gas with small scatterers

“…on the billiard table). Several powerful methods have been designed to prove bounds of the form (10), in particular using quasi-compactness of the transfer operator on Young towers [34] or anisotropic Banach spaces [14], coupling of standard pairs [10,Chapter 7] or most recently, Birkhoff cones [13]. However, each of these methods involve some non-constructive compactness argument which is the reason why there is no explicit information available on how the rate of decay (i.e.…”

“…Proof Let {W } ∈L be the foliation of the set {κ ρ = ξ +ξ N } into stable leaves. We can parametrise these leaves by their endpoints ( , π 2 ) in S 0 , then L is an interval of length c τ min < w (φ) < − 1 2π because of the direction of the stable cones, see (13). Let ν be a measure on L that produces the decomposition of Lebesgue measure m on {κ ρ = ξ + ξ N } along stable leaves.…”

Thermodynamic formalism for piecewise expanding maps in finite dimension

“…The geometric approach [15,30] has been used to study the SRB measure of piecewise hyperbolic maps with controlled complexity in dimension two ( [26]), but also the SRB measure and other equilibrium states of Sinai billiard maps and flows ( [28,9,7,8,6]). It has recently been extended to the random Lorentz gas, via Birkhoff cones [27]. Estimates on the essential spectral radius for micro-local spaces were 5 obtained ( [10,11]) for weighted piecewise hyperbolic surface maps.…”

“…Fix t ≥ 0. In Step I, we shall prove that27 sup µ∈Erg(T )h µ (T ) + log |G|dµ − tχ µ (DT ) ≤ log P * top ({log(G (n) ν t n )}) .InStep II, we shall find µ 0,t ∈ Erg(T ) withh µ0,t (T ) + log |G|dµ 0,t − tχ µ0,t (DT ) = log P * top ({log(G (n) ν t n )}) .Both steps will use Proposition 4.1.We can assume inf |G| > 0 by (57). We associate the sequence of continuous functions {log f π n,t } to {log f n,t = log(|G (n) |ν t n )} via (66).…”