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Projective Cones for Sequential Dispersing Billiards

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Abstract

We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential billiards (the billiard changes at each collision in a prescribed manner), open billiards (the dynamics exits some region or dies when hitting some obstacle) and many other examples. In particular, we include applications to chaotic scattering and the random Lorentz gas.

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Notes

  1. We do not claim that each such T is unique. It may be that \(T(Q) = T(Q')\) pointwise (consider a \(90^\circ \) rotation of a given configuration Q), yet for our purposes they will be considered distinct elements of \({\mathcal {F}}(\tau _*, {\mathcal {K}}_*, E_*)\).

  2. The abstract set-up in [DZ2] also allows billiard tables with infinite horizon and those subjected to external forces, but we are not concerned with the most general case here.

  3. Indeed, the distance \(\mathbb {d}\) allows configurations to move from finite to infinite horizon (see [DZ2, Section 6.2]), but we will not need that here as we will restrict ourselves to finite horizon configurations.

  4. The parameter \(\gamma \in (0,1)\) is from the cone condition (4.8).

  5. Closed here means that for all \(f,g \in {\mathcal {C}}\) and sequence \(\{\alpha _n\}\subset {\mathbb {R}}\) such that \(\lim _{n\rightarrow \infty }\alpha _n=\alpha \) and \(g+\alpha _n f\in {\mathcal {C}}\) for all \(n\in {\mathbb {N}}\) we have \(g+\alpha f\in {\mathcal {C}}\cup \{0\}\).

  6. By independent of the cone parameters, we mean that we may first fix \(\chi < 1\), and then choose \(c,A,L, \delta \) satisfying the conditions of Sect. 5.3 so that the contraction by \(\chi \) is obtained for all choices of \(c' > c\), \(A'>A\), \(L'>L\) and \(\delta ' < \delta \) that satisfy those conditions. Note, however, that larger \(A'>A\) requires \(n_0\) to increase in size.

  7. Note that \(\sum _i |T_nW_i| \le |W|\) and \( \frac{|T_nW_i|}{|W_i|}\le |J_{W_i}T_n|_{C^0(W_i)}\).

  8. Note that, by [CM, Proposition 4.47], given two maximal homogeneous subcurves of \(T_n^{-1}W^k\) that are connected by a vertical segment disjoint from \({\mathcal {S}}_{n}^{{\mathbb {H}}}\), there must exist two piecewise smooth curves in \({\mathcal {S}}_{n}^{{\mathbb {H}}}\) that connect the boundaries of such two subcurves forming a rectangle that does not contain any element of \({\mathcal {S}}_{n}^{{\mathbb {H}}}\) in its interior. Thus \(U^2_{i,+}\) must be a connected subcurve.

  9. Notice that since we subdivide curves in \({\mathcal {G}}_n^\delta (W)\) according to length \(\delta \) and not \(\delta _0\), the estimate of Lemma 3.3(b) becomes \({{\bar{C}}}_0 \delta ^{-1} |W| + C_0 \theta _1^n \le 2 {{\bar{C}}}_0 + C_0 \theta _1^n\), since \(|W| \le 2\delta \).

  10. Since the \(U^k_j\) are vertically matched, the term on the right hand side of (5.8) proportional to \(C_s\) is absent here.

  11. The second inequality in (6.2) follows from (3.7).

  12. Recall from Sect. 2.1 that by Proposition 2.2, \({\mathcal {Q}}(Q_{\iota _j}, E_*; \kappa ) = \{ Q \in {\mathcal {Q}}(\frac{1}{2} \tau _*, \frac{1}{2} {\mathcal {K}}_*, E_*) : \mathbb {d}(Q, Q_{\iota _j}) < \kappa \}\).

  13. Although the stable/unstable directions in M vary, they always belong to the global stable/unstable cones defined in (H1) and so are uniformly transverse.

  14. The choice of \(\delta _0^4\) will be needed in Lemma 6.7.

  15. Recall that \({{\bar{C}}}_0\) is from Lemma 3.3.

  16. These Cantor rectangles \(R_i\) are maximal in the sense that they are the intersection of the maximal families of local invariant manifolds \({\mathfrak {S}}^{s/u}(D(R_i))\) that fully cross the solid rectangle \(D(R_i)\).

  17. Indeed, using Proposition 5.1 and choosing \(L \ge 60\), we can always choose \(\chi = \frac{8}{9}\), although this will affect the choice of \(N_{\mathcal {F}}\).

  18. Recall that \(n_*\) is defined in Lemma 6.6 while \(N(\delta )^-\) is defined in Eq. (6.4).

  19. A semi-norm \(\Vert \cdot \Vert \) is order preserving if \(-g\preceq f\preceq g\) implies \(\Vert f\Vert \le \Vert g\Vert \).

  20. [LSV, Lemma 2.2] is stated for order preserving norms but its proof holds verbatim for order preserving semi-norms, see [DKL1, Lemma D.4].

  21. According to (O1), \(W^k\) is divided into at most \(P_0\) pieces, and \(W^k \cap H^c\) comprises at most \(\frac{P_0}{2}+1\) of them. Each such piece can give rise to at most 2 unmatched pieces.

  22. Since \(I_{W^1_j} = I_{W^2_j}\), the term on the right side of (5.8) proportional to \(C_s\) is absent in this case.

  23. Since we have fixed the cone constants c, A, L, the number \({{\bar{n}}}_\delta \) depends on the constants appearing in (O\(1'\)) and (O2) as well as \(\mu _{\tiny {\text{ SRB }}}(H)\) and the choice of \(\delta \), from Lemma 8.6.

  24. \(N_{{\mathcal {F}}}'\) is number from Theorem 6.12 applied to the cone with larger constants \(c'', A'', L''\).

  25. Requiring \(\mu _{\tiny {\text{ SRB }}}(H) \le 1/2\) enables a uniform choice of \(n_\star \) for all \(H \in {\mathcal {H}}(P_0, C_t)\).

  26. Remember that the non-eclipsing condition is the requirement that the convex hull of any two obstacles does not intersect any other obstacle.

  27. Recall that \(\varphi \in [-\frac{\pi }{2},\frac{\pi }{2}]\) is the angle made by the post-collision velocity vector and the outward pointing normal to the boundary.

  28. This claim implies that the cone is Archimedean.

  29. Note that \(\mathring{{\mathcal {L}}}_{S}\) extends naturally to \(({\mathcal {C}}^1(M))'\) and therefore to \({\mathcal {C}}_\star \).

  30. If instead \(f \in {\mathcal {C}}_{c,A,L}(\delta )\), \(f \ge 0\) and \(\int f \, d\mu _{\tiny {\text{ SRB }}}=1\), then \(\Vert f \Vert _{C^1}\) can be dropped from the right hand side.

  31. The assumption that all obstacles are circular is not essential and can be relaxed by requiring that the obstacles at the corners are symmetric with respect to reflections as described in Sect. 8.4.

  32. Finite horizon requires \(r \ge \frac{1}{1+\sqrt{2}}\), yet our added condition that a particle cannot cross diagonally from, say, \({{\hat{R}}}_1\) to \({{\hat{R}}}_2\) without making a collision requires further that \(r \ge \frac{1}{3}\).

  33. The hole depends on the trajectory of x, which is different in different cells and hence depends on z, while the gates \({{\hat{R}}}_i\) are independent of z.

  34. By (mod 4*) we mean cyclic addition on 1, 2, 3, 4 rather than 0, 1, 2, 3.

  35. Since \(z_0 = (0,0)\), it is equivalent to specify \(z_1, \ldots z_n\) or \(w_{k_0}, \ldots w_{k_{n-1}}\) since \(w_{k_j}\) can be recovered as \(w_{k_j} = z_{j+1} - z_j\).

  36. Here in fact our operators are of the form \({\mathcal {L}}^n\mathbb {1}_H\) while in Proposition 8.7 they have the form \({\mathcal {L}}_n \mathbb {1}_{H^c}\) for some set H. Yet, this is immaterial since the boundaries of H and \(H^c\) in M are the same so that (O\(1'\)) and (O2), and in particular Lemma 8.6, apply equally well to both sets.

  37. Remark that [AL, Theorem 6.4] requires \(\mu _{\tiny {\text{ SRB }}}(G_i(z))\) to be the same for each i and z, independently of \(\omega \). This is precisely the case here since \(G_i(z)\) is defined as the projection of \({{\hat{R}}}_i\) under the inverse flow \(\Phi ^z_{-t}\), and Leb\(({{\hat{R}}}_i \times [0, 2\pi ))\) in the phase space of the flow is independent of i, while \(\mu _{\tiny {\text{ SRB }}}\) is the projection onto M of Lebesgue measure, which is invariant under the flow.

  38. The arguments in [AL, Section 6] are developed for expanding maps, but the relevant parts apply verbatim to the present context.

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Acknowledgements

This work was supported by the PRIN Grant “Regular and stochastic behavior in dynamical systems" (PRIN 2017S35EHN). C. Liverani acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. M. Demers was partially supported by NSF grants DMS 1800321 and DMS 2055070. C. Liverani is a member of the Editorial Board of Communications in Mathematical Physics.

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Authors and Affiliations

  1. Department of Mathematics, Fairfield University, 06824, Fairfield, CT, USA

    Mark F. Demers

  2. Dipartimento di Matematica, Università di Roma II, Tor Vergata, 00133, Rome, Italy

    Carlangelo Liverani

  3. GNFM-INDAM, Rome, Italy

    Carlangelo Liverani

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  1. Mark F. Demers
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Correspondence to Mark F. Demers.

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Communicated by S. Dyatlov.

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We thank Jochen Broecker, Tobias Kuna and especially Lea Oljaca for pointing out to us the possibility to prove Lemma 4.10. This work was supported by the PRIN Grant “Regular and stochastic behaviour in dynamical systems” (PRIN 2017S35EHN). C.L. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. M.D. was partially supported by NSF grants DMS 1800321 and DMS 2055070.

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Demers, M.F., Liverani, C. Projective Cones for Sequential Dispersing Billiards. Commun. Math. Phys. 401, 841–923 (2023). https://doi.org/10.1007/s00220-023-04657-1

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