Explore the vast range of titles available.

We prove a conjecture on the second minimal odd periodic orbits with respect to Sharkovski ordering for the continuous endomorphisms on the real line. A (2k+1)-periodic orbit β1<β2<⋯<β2k+1, (k≥3) is called second minimal for the map f, if 2k−1 is a minimal period of f|[β1,β2k+1] in the Sharkovski ordering. Full classification of second minimal orbits is presented in terms of cyclic permutations and directed graphs of transitions. It is proved that second minimal odd orbits either have a Stefan-type structure like minimal odd orbits or one of the 4k−3 types, each characterized with unique cyclic permutations and directed graphs of transitions with an accuracy up to the inverses. The new concept of second minimal orbits and its classification have an important application towards an understanding of the universal structure of the distribution of the periodic windows in the bifurcation diagram generated by the chaotic dynamics of nonlinear maps on the interval.

A discrete faithful representation of the free group on gg generators Fg{F_g} into Isom+⁡(H3)\\operatorname {Isom}_ + ({{\\mathbf {H}}^3}) is said to be a Schottky group if (H3∪DΓ)/Γ({{\\mathbf {H}}^3} \\cup {D_\\Gamma })/\\Gamma is homeomorphic to a handlebody Hg{H_g} (where DΓ{D_\\Gamma } is the domain of discontinuity for Γ\\Gamma’s action on the sphere at infinity for H3{{\\mathbf {H}}^3}). Schottky space Sg{\\mathcal {S}_g}, the space of all Schottky groups, is parameterized by the quotient of the Teichmüller space T(Sg)\\mathcal {T}({S_g}) of the closed surface of genus gg by Mod0(Hg){\\operatorname {Mod} _0}({H_g}) where Mod0(Hg){\\operatorname {Mod} _0}({H_g}) is the group of (isotopy classes of) homeomorphisms of Sg{S_g} which extend to homeomorphisms of Hg{H_g} which are homotopic to the identity. Masur exhibited a domain O(Hg)\\mathcal {O}({H_g}) of discontinuity for Mod0(Hg){\\operatorname {Mod} _0}({H_g})’s action on PL(Sg)PL({S_g}) (the space of projective measured laminations on Sg{S_g}), so B(Hg)=O(Hg)/Mod0(Hg)\\mathcal {B}({H_g}) = \\mathcal {O}({H_g})/{\\operatorname {Mod} _0}({H_g}) may be appended to Sg{\\mathcal {S}_g} as a boundary. Thurston conjectured that if a sequence {ρi:Fg→Isom+⁡(H3)}\\{ {\\rho _i}:{F_g} \\to \\operatorname {Isom}_ + ({{\\mathbf {H}}^3})\\} of Schottky groups converged into B(Hg)\\mathcal {B}({H_g}), then it converged as a sequence of representations, up to subsequence and conjugation. In this paper, we prove Thurston’s conjecture in the case where Hg{H_g} is homeomorphic to S×IS \\times I and the length lNi((∂S)∗){l_{{N_i}}}({(\\partial S)^\\ast }) in Ni=H3/ρi(Fg){N_i} = {{\\mathbf {H}}^3}/{\\rho _i}({F_g}) of the closed geodesic(s) in the homotopy class of the boundary of SS is bounded above by some constant KK.

We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local k-algebra R, not necessarily commutative, where k is a field. The class of topological modules we consider include all those of finite rank over k and some of infinite rank as well, namely those with a Schauder basis in the sense of$\\S1$. This generalizes the results of [Sh], where the result was obtained in a different way in case the ring R is the completion of the local ring of a plane curve singularity and the module is kn. Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called \"growth functions\" to handle explicit episilonics involving the convergence of formal power series in noncommuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of [Sch] which were used in [Sh] and is more conceptual.

A weighted endomorphism of a Banach algebra is an endomorphism followed by a multiplier. In this paper we characterize compact weighted endomorphisms on the Banach algebra AC[a, b] of absolutely continuous functions on [a, b]. Moreover, we apply the obtained characterization to describe the spectra of such operators on AC[a, b].

The theme of the paper is the question of existence and basic structure of transfer operators for endomorphisms of a unital C*-algebra. We establish a complete description of non-degenerate transfer operators, characterize complete transfer operators and clarify their role in crossed product constructions. Also, we give necessary and sufficient conditions for existence of transfer operators for commutative systems, and discuss their form for endomorphisms of B(H) which is relevant to the Kadison-Singer problem.