The bifurcation of C1-continuous families of maps of the interval or circle is studied. It is shown, for example, that period-tripling cannot occur. This yields topological properties of the stratification of C1(I, I) induced by the Sarkovskii order, and corresponding bifurcation properties.

Relationships involving homoclinic orbits of various periods and the Sarkovskii stratification are given and corresponding bifurcation properties are derived. It is shown that if a continuous map has one homoclinic periodic orbit, it has infinitely many. In any family of C1 maps going from zero to positive entropy, infinitely many homoclinic bifurcations occur, involving periods which are successively smaller powers of two.

Let

(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

It is shown that any group isomorphism between the Ck diffeomorphism groups of two finite-dimensional, boundaryless paracompact manifolds is induced by a Ck diffeomorphism between the manifolds (1 ≤ k ≤ ∞).

In response to a question of Newhouse, we show that in many cases an Anosov automorphism can be uniformly approximated by a Smale diffeomorphism of the same topological entropy. We conjecture this can be done whenever the automorphism is sufficiently hyperbolic.

A three-dimensional example is given that suggests this approximation is not always possible.

A recent result of J. Mather [1] about the existence of quasi-periodic orbits for twist maps is derived from an appropriately modified version of G. D. Birkhoff's classical theorem concerning periodic orbits. A proof of Birkhoff's theorem is given using a simplified geometric version of Mather's arguments. Additional properties of Mather's invariant sets are discussed.

Let SA be an irreducible and aperiodic topological Markov chain. If SĀ is an irreducible and aperiodic topological Markov chain, whose topological entropy is less than that of SA, then there exists an irreducible and aperiodic topological Markov chain, whose topological entropy equals the topological entropy at SĀ, and that is a subsystem of SA. If S is an expansive homeomorphism of the Cantor discontinuum, whose topological entropy is less than that of SA, and such that for every j∈ℕ the number of periodic points of least period j of S is less than or equal to the number of periodic points of least period j of SA, then S is topological conjugate to a subsystem of SA.

We give a proof of Pesin entropy formula in a very general setting.

There is a theorem of Sarkovskii which characterizes the sets of periods of periodic points for continuous maps of an interval into itself. We give a similar characterization for continuous maps of degree one of a circle into itself.

We consider perturbations of certain transitive maps of an interval into itself and estimate how far from the transitivity the perturbed maps are. The distance turns out not to be of greater order than the square of the size of the perturbation.

Consider an analytic differential equation ο = adx + bdy with an algebraically isolated singularity and without a separatrix. The germ at 0∈ℝ2 of the 1-jet y dy is either a focus or a centre. The equation has C∞ normal form of the type = with F(x) = F(−x) if and only if the germ is a centre.