In preparation for these remarks, I looked up Carmen's papers on MathSciNet and discovered he has more than three times as many as I do. Of course, every teacher is proud when his student surpasses him, and I am indeed proud of Carmen.

I noticed that his first published paper was a counter-example to a conjecture of mine. In view of the oft used metaphor of the thesis adviser as father, this brings to mind the story of Oedipus. I searched the web for a suitable account of this story and came up with Tom Lehrer's famous song. This version of the story emphasizes more the love of Oedipus for his mother which seems to me to be irrelevant here. However, I also found the following summary of the story which I want to share with you.

Oedipus

     Higgledy piggledy
        Oedipus Tyrannos
        murdered his father, used
        mama for sex.
     This mad debauch, not so
        incomprehensibly,
        left poor Jocasta and
        Oedipus Wrecks.        -Joan Munkacsi
    

Now this verse is an example of the so-called "double dactyl" form. A dactyl is a three syllable poetic foot with the accent on the first syllable, e.g. "Higgledy". Thus a double dactyl is two dactyls in a row like "Higgledy piggledy". I found the following definition on the web:

A double dactyl is also a poem, a form invented by Anthony Hecht and Paul Pascal. Quite like a limerick, it has a rigid (if peculiar) structure. Two stanzas, each comprising three lines of dactylic dimeter followed by a line with a dactyl and a single accent. The two stanzas have to rhyme on their last line.

Double Dactyl

     Long short short, long short short
        Dactyls in dimeter,
        Verse form with choriambs
        (Masculine rhyme):
     One sentence (two stanzas)
        Hexasyllabically
        Challenges poets who
        Don't have the time.   - Roger L. Robison
    

Here is an example which may remind you of my talk this morning.

Boring Colloquia

     Higgeldy piggeldy
        Boring colloquia:
        Incomprehensible 
        Gobbledy-gook.
     Second to nothing in
        Soporificity,
        Except for, possibly,
        Reading the book.       -Robin Pemantle (1994)
    

I wrote a double dactyl commemorating Carmen's Oedipus paper. To help you understand the poem I need to supply some background. Anosov himself proved that his diffeomorphisms are ergodic. The proof is difficult. I noticed the following argument. A dynamical system ƒ is ergodic iff the only ƒ-invariant functions are constant, and this would follow if the only ƒ-invariant one form is zero, or dually, if the vector fields in the range of (1-ƒ*) are dense. Of course, the functions should be square integrable and the exterior derivative takes the Hilbert space of square integrable functions to the Sobolev space of one forms with a square integrable antiderivative, i.e. the dual space of the Sobolev space of vector fields with one square integrable derivative. By a theorem of Mather, ƒ is Anosov if and only of the operator (1-ƒ*) is an isomorphism on the space of continuous vector fields, and it seemed not unreasonable to hope that it might at least have dense range on the Sobolev space. The idea is so simple, it must be correct.

Let us call the desired property "infinitesimal H-ergodicity" -- the letter H is added to remind that the appropriate space of vector fields is a Sobolev space. Anosov and Arnold called the equation (1-ƒ*)ξ=η the "homological" equation so we call the operator (1-ƒ*) the homological operator. It is reasonable to call a vector field in the range of (1-ƒ*) a "derived" vector field. For reasons of poetic euphony we need to insert the prefix "co". Carmen's example is a four dimensional toral automorphism and he uses Fourier series and the characteristic (eigen) values of the defining matrix to do the analysis. Finally, just as I often have to explain to people that my last name is "Robbin" not "Robbins", Carmen often has to explain that his last name is pronounced "Chi-co-ne" not "Chi-cone". His name must be pronounced correctly to achieve the double dactyl form in the poem. The title of my poem is the same as the title of Carmen's paper.

Anosov does not imply infinitesimal ergodicity

    Infinitesimal
       H-Ergodicity
       Seems an immediate
       Consequence of
    Cohomological
       Hyperbolicity
       Something Anosov has
       Taught us to love.
       
    Carmen Chicone said
       This is  baloney, the
       Simp-le-st example
       Indicates hence
    Characteristically
       Fourieristically
       Coderived vector fields 
       Need not be dense.
   

Happy birthday Carmen!