43,832 research outputs found
On the uniqueness of the coincidence index on orientable differentiable manifolds
The fixed point index of topological fixed point theory is a well studied
integer-valued algebraic invariant of a mapping which can be characterized by a
small set of axioms. The coincidence index is an extension of the concept to
topological (Nielsen) coincidence theory. We demonstrate that three natural
axioms are sufficient to characterize the coincidence index in the setting of
continuous mappings on oriented differentiable manifolds, the most common
setting for Nielsen coincidence theory.Comment: Major addition- section added at end. Previous material mostly
unchanged. Numbering, etc. now in sync with publication versio
Linearization models for parabolic dynamical systems via Abel's functional equation
We study linearization models for continuous one-parameter semigroups of
parabolic type. In particular, we introduce new limit schemes to obtain
solutions of Abel's functional equation and to study asymptotic behavior of
such semigroups. The crucial point is that these solutions are univalent
functions convex in one direction. In a parallel direction, we find analytic
conditions which determine certain geometric properties of those functions,
such as the location of their images in either a half-plane or a strip, and
their containing either a half-plane or a strip. In the context of semigroup
theory these geometric questions may be interpreted as follows: is a given
one-parameter continuous semigroup either an outer or an inner conjugate of a
group of automorphisms? In other words, the problem is finding a fractional
linear model of the semigroup which is defined by a group of automorphisms of
the open unit disk. Our results enable us to establish some new important
analytic and geometric characteristics of the asymptotic behavior of
one-parameter continuous semigroups of holomorphic mappings, as well as to
study the problem of existence of a backward flow invariant domain and its
geometry
The Entropy of Co-Compact Open Covers
Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space(compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen's entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces, too
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
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