Continuous Bijection Research Articles

WHEN ARE CONTINUOUS ISOTONE BIJECTIONS ORDER AUTOMORPHISMS?

The purpose of this paper is to study weakly continuous isotone bijections on a lattice of residuated maps (e.g. the space of probability distribution functions). We investigate the connection between weakly continuous isotone bijections and order automorphisms. Our final result gives necessary and sufficient conditions for a continuous isotone bijection to be an order automorphism.

Symmetry groups of Markov processes and the diagonal principle

LetXt be a transient Markov process onE with potentialU and excessive measures Exc. Let Sym be the group of finely continuous bijections onE which leave Exc invariant. From each subgroupJ of Sym, we construct the spaceF ofJ orbits inE and the canonical surjection Ψ:E→F. We show that Ψ(Z1) is a strong Markov process onF, whereZt is obtained fromXt by anh transform and time change. IfH is a subgroup of Sym withHe={ϕ∈H: ϕ(e)=e} trivial, thenH induces a group structure onE. There is a function 0<h<∞ such thatV(x, ·)=h(x)−1 U(x, ·) is translation-invariant onE; h need not be excessive. Analogous results for supermartingale multiplicative functionals are given, and a general framework is proposed.

On simultaneous Abel equations

We investigate the continuous and measurable solutions of the system of Abel's functional equations (1) $$\begin{gathered} \varphi (f(x)) = \varphi (x) + 1 \hfill \\ ,x \in (a,b) \hfill \\ \varphi (g(x)) = \varphi (x) + s \hfill \\ \end{gathered} $$ and their applications to the iteration theory. Let us assume the following hypothesis: (H)f, g: (a, b) → (a, b) are continuous bijections andf o g = g o f. Moreoverf n (x) ≠ g m (x) for everyx∈(a, b) and everyn, m ∈ ℤ such that |n| + |m| ≠ 0. LetL be the set of limit points of {f n o g m (x): n, m ∈ ℤ}, wherex ∈(a, b) (L does not depend ofx). The setL is either a perfect and nowhere dense set orL = 〈a, b〉. Theorem.If f and g satisfy hypothesis (H), then there is a unique s ∈ ℝ such that the system (1) has a continuous solution. For this s the system (1) has a continuous solution ϕ unique up to an additive constant. This solution ϕ is monotonic, ϕ[L ⋂ (a, b)] = ℝ and s is irrational. Moreover ϕ is invertible if and only if L = 〈a, b〉. Corollary.Let f and g satisfy hypothesis (H). Then there exists a continuous iteration group {f t t} such that ft 1 =f and g ∈ {f t } if and only if L = 〈a, b〉. Moreover this iteration group is unique. Further let us assume the following hypothesis: (C)f, g: (a, b) →(a, b) are continuous bijections such thatf(x) ≠ x, g(x) ≠ x forx ∈(a, b), fog=gof andf n =g m for somen, m ∈ ℤ\{0}. Theorem.Let hypothesis (C) be fulfilled. Then the system (1) for s = n/m has a continuous and invertible solution. This solution depends on an arbitrary function. Corollary.Let f and g satisfy hypothesis (C), then there exist infinitely many continuous iteration groups {f t } such that f 1 =f and g ∈ {f t }.

On inverses of almost continuous bijections

On inverses of almost continuous bijections

On Minkowski’s singular function

The function f on the closed unit interval I to be considered here was introduced by Minkowski. It maps the algebraic irrationalities of degree at most 2 continuously to the rationals in I. If z = a/b and z' = c/d are (reduced) fractions, then we write z * z' for the mediant (a + c)/(b + d) and z z' for the arithmetic mean. Let Q denote the (dense) set of rationals in I with terminating dyadic expansion, i.e. the set of numbers x= n2'' (n = 0,.. .,2-1, v E N). Define a function g on Q inductively as follows: Let y' = 0, y= 1. Suppose the values yn = g(Xn) have been defined on level v. Then put y2+' = Yk (0 < k < 2W) and insert +1 =YV1 *Yk (1< k <2W) to obtain the images y+' = g(x+') on level v+ 1. The resulting function g is a strictly increasing continuous bijection from Q to the rationals. Note that g is constructed so as to have the property g(x) *g(x') = g(x x') whenever x, x' E Q, x' x = 21-, v E N. The inverse of g can be extended to the reals to yield a strictly increasing continuous distribution function f inducing a (singular) probability measure on I. We will be concerned with the regular and semiregular continued fraction expansions y=J +l 3+ [= la),...],l ,,... E N and y = a-i b= [a, b, ... .1, a, b, ... E N {1}. In the dyadic expansion of elements x E I we collect 0's and l's and write x = 0, -110O1 60, , letting a 1,/,-y,6, ,. . . denote string lengths (a, 3, -y, 6, E, ... E N). The following has been proved by different arguments (see [2] for references): (i) a+1;2i_j,'-y+2,26_1,E+2, ... f'(011 ,0,_1 16 0, ) (this is [2, (5)]); (ii) f is singular. Our purpose here is to present a simple unified approach to both properties by splitting up (i) into the following equalities, the second one making (ii) evident:

Continuous bijections on manifolds

AbstractThe main results of the paper give necessary and sufficient conditions as well as sufficient conditions that continuous bijections of a manifold onto itself be homeomorphisms. Such conditions include the embedding of manifolds, preserving ends, preserving closed half-rays and restrictions on boundary components. A number of counterexamples are given to likely conjectures.

A Continuous Linear Bijection with Discontinuous Inverse

(1971). A Continuous Linear Bijection with Discontinuous Inverse. The American Mathematical Monthly: Vol. 78, No. 5, pp. 532-533.

A Continuous Linear Bijection with Discontinuous Inverse

A Continuous Linear Bijection with Discontinuous Inverse