Abstract:Abstract.A one parameter family of piecewise linear measure preserving transformations of a torus which can be viewed as a perturbation of the twist mapping is introduced. Theorems on their ergodic properties for an infinite set of parameters are proved. For some parameters coexistence of stochastic and integrable behaviour is obtained.
“…The nonsmooth invariant curves form barriers to the existence of dense orbits within the exceptional set. There has been some work on piecewise versions of the standard area preserving map; for example, see [9,11,31]. In certain cases these can be reduced to piecewise isometries [4]; we note that only discontinuous piecewise isometries can have nontrivial dynamics.…”
Abstract. Invertible piecewise isometric maps (PWIs) of the plane, in spite of their apparent simplicity, can show a remarkable number of dynamical features analogous to those found in nonlinear smooth area preserving maps. There is a natural partition of the phase space into an exceptional set, E, consisting of the closure of the set of points whose orbits accumulate on discontinuities of the map, and its complement. In this paper we examine a family of noninvertible PWIs on the plane that consist of rotations on each of four atoms, each of which is a quadrant. We show that this family gives examples of global attractors with a variety of geometric structures. On some of these attractors, there appear to be nonsmooth invariant curves within E that form barriers to ergodicity of any invariant measure supported on E. These invariant curves are observed to appear on perturbations of an "integrable" case where the exceptional set is a union of annuli and it decomposes into a one-dimensional family of interval exchange maps that may be minimal but nonergodic. We have no adequate theoretical explanation for the curves in the nonsmooth case, but they appear to come into existence at the same times as an explosion of periodic islands near where the interval exchanges used to be located. We exhibit another example-a piecewise rotation on the plane with two atoms that also appears to have nonsmooth invariant curves.
“…The nonsmooth invariant curves form barriers to the existence of dense orbits within the exceptional set. There has been some work on piecewise versions of the standard area preserving map; for example, see [9,11,31]. In certain cases these can be reduced to piecewise isometries [4]; we note that only discontinuous piecewise isometries can have nontrivial dynamics.…”
Abstract. Invertible piecewise isometric maps (PWIs) of the plane, in spite of their apparent simplicity, can show a remarkable number of dynamical features analogous to those found in nonlinear smooth area preserving maps. There is a natural partition of the phase space into an exceptional set, E, consisting of the closure of the set of points whose orbits accumulate on discontinuities of the map, and its complement. In this paper we examine a family of noninvertible PWIs on the plane that consist of rotations on each of four atoms, each of which is a quadrant. We show that this family gives examples of global attractors with a variety of geometric structures. On some of these attractors, there appear to be nonsmooth invariant curves within E that form barriers to ergodicity of any invariant measure supported on E. These invariant curves are observed to appear on perturbations of an "integrable" case where the exceptional set is a union of annuli and it decomposes into a one-dimensional family of interval exchange maps that may be minimal but nonergodic. We have no adequate theoretical explanation for the curves in the nonsmooth case, but they appear to come into existence at the same times as an explosion of periodic islands near where the interval exchanges used to be located. We exhibit another example-a piecewise rotation on the plane with two atoms that also appears to have nonsmooth invariant curves.
“…The present paper is the continuation of the paper [2] in which only large perturbations were considered. The reader is advised to consult [2] for more motivations and references.…”
Section: Wojtkowskimentioning
confidence: 96%
“…Both Fi and F 2 preserve the Lebesgue measure d<f> i d<f> 2 . We shall study transformations of the form …”
Section: Fl{((> 1 Z) = ( 1+ 2 2)mentioning
confidence: 99%
“…It was proved in [2] that, for A = B, T has strong mixing properties in the whole torus if A > 4 and in some invariant domains for a sequence of parameters A from the interval [1,4]. So A was quite far from zero.…”
Section: Jomentioning
confidence: 99%
“…For the sake of completeness, we repeat the definition of almost hyperbolicity from [2]. DEFINITION and T"'\ x \ is Bernoulli (see [4] and references in it).…”
Section: Almost Hyperbolicity Of T In 3>mentioning
Dedicated to the memory of V. M. AlexeyevAbstract. It is proved that for a sequence of arbitrarily small piecewise linear perturbations of the twist map, there is a domain with stochastic behaviour (almost hyperbolicity). The measure of this domain has the asymptotics uA In-^-(1+0 (1)
), A->0A where A is the magnitude of the perturbation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.