Three bands, not one sign
A sign covers thirty degrees, and the tradition has divided each of them into three ten degree faces for a very long time. The system predates much of what surrounds it and comes originally from Egyptian star lists, where the thirty six decans marked the hours of the night.
Capricorn's first decan runs from zero to ten degrees, the second from ten to twenty, the third from twenty to thirty. A birth on 3 January places the Sun between roughly 12.0 and 13.4 degrees, taking noon positions from 1930 to 2035, which puts it in the second band in every one of those years.
The reason the position holds so steadily is the leap year rule. The Sun reaches a given longitude about six hours later each year than the last, and the extra February day cancels the accumulated slip every fourth year, so the degree oscillates in a narrow band rather than drifting.
Who rules the band, and why the answer varies
Two rulership schemes are in common use and they disagree about this decan. The Chaldean order runs Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter repeatedly through all thirty six faces from the start of Aries, and the count lands Mars on the second decan of Capricorn. The triplicity scheme instead gives each decan of a sign to a sign of the same element in sequence, so the second earth face here goes to Taurus, and therefore to Venus.
Neither is a corruption of the other. They answer different questions, and popular articles often quote one without naming it, which is the usual source of confusion when two sites give a reader different rulers for the same birthday.
The Chaldean order is not arbitrary either. It is the same sequence that produces the planetary hours and, through them, the names of the days of the week in several languages, which is one reason it has survived intact for so long.
What is read into it
The tradition treats Capricorn as cardinal earth under Saturn, patient and structural, and treats the decan as a colouring rather than a second sign. With Mars attached, older texts describe the band as Saturn's endurance with a sharper edge on it, a readiness to push where the first decan would wait. With Taurus attached, the same ten degrees read as more settled and more physical.
Both readings are interpretation. What is measurable is the position: the sign, the degree and the band. What is inherited is the meaning, and the two are worth keeping apart.
Everything the decan does not cover
Knowing the decan sharpens the description of one placement. It says nothing about the rest of the chart. The Moon shifts about thirteen degrees a day, so the hour of a 3 January birth decides whether it is in one sign or the next. The rising sign changes roughly every two hours and depends on the birthplace as well as the clock.
The widget above computes your solar degree from your own year and location, names the decan it falls in, and tells you how far your Sun sits from the nearest sign boundary.
Two people born hours apart on the same 3 January therefore share a Sun to within a fraction of a degree and can share very little else. That gap between what the date fixes and what it leaves open is the whole reason a birth time is asked for.
Questions people ask about this
What decan is 3 January?
The second decan of Capricorn, ten to twenty degrees. The Sun sits near twelve degrees on this date, so the answer holds for every year in the modern calendar.
What is a decan?
A ten degree division of a sign. Each sign has three, giving thirty six across the zodiac. The scheme comes from Egyptian star lists and is older than several of the ideas layered on top of it.
Why do sites give different decan rulers?
Because the Chaldean order and the triplicity scheme produce different answers for most decans, and articles rarely say which one they are using. For this band the answers are Mars and Venus respectively.
What sign is 3 January?
Capricorn, with no year in which that changes. The date is a fortnight clear of the Sagittarius boundary and more than a fortnight clear of the Aquarius one.
Does the decan change my sun sign?
No. A decan is a subdivision inside the sign, so it refines the description rather than replacing it. A Capricorn in the second decan is fully Capricorn.
Is the decan more important than the sign?
Neither ranks above the other, and both are far less individual than the Moon and the rising sign. The decan is a refinement of a placement that everyone born in the same month shares.