153 research outputs found
Code & order in polygonal billiards
Two polygons are code equivalent if there are billiard orbits
which hit the same sequence of sides and such that the projections of the
orbits are dense in the boundaries . Our main results
show when code equivalent polygons have the same angles, resp. are similar,
resp. affinely similar
Homotopical rigidity of polygonal billiards
Consider two -gons and . We say that the billiard flows in and
are homotopically equivalent if the set of conjugacy classes in the
fundamental group of which contain a periodic billiard orbit agrees with
the analogous set for . We study this equivalence relationship and compare
it to the equivalence relations, order equivalence and code equivalence,
introduced in \cite{BT1,BT2}. In particular we show if is a rational
polygon, and is homotopically equivalent to , then and are
similar, or affinely similar if all sides of are vertical and horizontal
Constant slope maps and the Vere-Jones classification
We study continuous countably piecewise monotone interval maps, and formulate
conditions under which these are conjugate to maps of constant slope,
particularly when this slope is given by the topological entropy of the map. We
confine our investigation to the Markov case and phrase our conditions in the
terminology of the Vere-Jones classification of infinite matrices.Comment: 33 page
Does a billiard orbit determine its (polygonal) table?
We introduce a new equivalence relation on the set of all polygonal
billiards. We say that two billiards (or polygons) are order equivalent if each
of the billiards has an orbit whose footpoints are dense in the boundary and
the two sequences of footpoints of these orbits have the same combinatorial
order. We study this equivalence relation with additional regularity conditions
on the orbit
The topological entropy of Banach spaces
We investigate some properties of (universal) Banach spaces of real functions
in the context of topological entropy. Among other things, we show that any
subspace of which is isometrically isomorphic to contains a
functions with infinite topological entropy. Also, for any ,
we construct a (one-dimensional) Banach space in which any nonzero function has
topological entropy equal to .Comment: The paper is going to appear at Journal of Difference Equations and
Application
Monotony of solutions of some difference and differential equations
AbstractIn this contribution motivated by some analysis of the first author concerning bounds of topological entropy it is shown that a well known sufficient condition for a difference and differential equation with constant real coefficients to possess strictly monotone solution appears to be also necessary. Transparent proofs of adequate generalizations to Banach space analogs are presented
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