Asymptotics of Dispersing Billiards with Holes

Definitions and mathematical statement

Let $T\colon M\dashrightarrow M$ denote the finite-horizon dispersing billiard map in the supplied record, where $M$ is its collision phase space, and let $H\subseteq M$ be a hole: a trajectory escapes when it enters $H$. Finite horizon means that the lengths of free flights between successive collisions are uniformly bounded. The record does not specify the billiard geometry, dimension, treatment of singular trajectories, or admissible holes and initial distributions.

To display the ambiguity in the evolution of distributions, suppose that $M$ has a measurable structure, $H$ is measurable, and $\mu_0$ is a probability measure for which the required iterates are defined almost everywhere. Under the explicit convention that escape is checked after each iterate, set \[ \tau_H(x)=\inf\{k\geq 1:T^k x\in H\},\qquad \inf\varnothing=\infty. \] For a measurable set $A\subseteq M$ and an integer $n\geq0$, define the surviving, unnormalized distribution by \[ \nu_n(A)=\mu_0\left(\{x:\tau_H(x)>n,\ T^n x\in A\}\right). \] Its mass is $s_n=\nu_n(M)=\mu_0(\tau_H>n)$. Whenever $s_n>0$, the distribution conditioned on survival is \[ \widehat\nu_n(A)=\frac{\nu_n(A)}{s_n}. \] These formulas give possible meanings of the source's $\mu_n$; the supplied record does not select one. Checking escape at time zero is another convention and requires corresponding changes if $\mu_0(H)>0$. The measurable framework above is explanatory, not a claim that the source supplies the necessary hypotheses.

Asymptotic question. For a finite-horizon dispersing billiard map with a hole $H$, determine the behavior as $n\to\infty$ of the evolved initial distribution $\mu_n$. A precise theorem or conjecture cannot be extracted without specifying the admissible billiards, holes and initial distributions, the escape and normalization conventions, and the intended notion of asymptotic behavior or convergence. In particular, the record does not determine whether the target concerns $\nu_n$, its surviving mass, or $\widehat\nu_n$, and supplies no particular proposed limit.

Short English statement

The question asks what happens over time to a distribution of billiard trajectories when trajectories entering a hole are removed. The remaining total mass and the distribution conditioned on survival describe different aspects of this evolution. The supplied statement leaves their intended role, the admissible initial distributions and holes, and the meaning of convergence unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001293$, title Asymptotics of Dispersing Billiards with Holes, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied source citation: Dynamische Systeme (2010), Oberwolfach Reports, pp. 1787--1842, DOI 10.4171/owr/2009/32. Source URL and literature-source link labelled "OWR report": https://doi.org/10.4171/owr/2009/32. The record describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2010 designation and its 2009 DOI component are retained as supplied.
• Supplied literature-source link labelled "Demers--Wright--Young on billiards with holes": https://doi.org/10.1007/s00220-009-0941-y. The supplied assessment attributes to Demers, Wright, and Young limiting-distribution results for a broad billiard class with small holes, described as a partial answer to the extracted asymptotic question. Exact theorem hypotheses and bibliographic details beyond this attribution and link are not supplied.
• Literature assessment checked at 2026-08-21: the supplied status is partially_solved. This is supplied dated curation derived from the OWR record, not independently verified here; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification note records only the addition of mathematical delimiters without a change of mathematical content. It does not establish mathematical completeness or correctness.